INFORMATION TO USERS
This manuscript has been reproduced from the microfilm master. UMI fihns the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter 6ce, while others may be from any type of computer printer.
The quality of this reproduction Is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedtfarough, substandard margins, and improper alignment can adversely affect reproduction.
In the unlikely event that the author did not said UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, b^inning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book.
Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Infiumadon Company 300 North Zeeb Road, Ann Aibor MI 48106-1346 USA 313/761-4700 800/521-0600
A CENTRAL LIMIT THEOREM FOR COMPLEX-VALUED PROBABILITIES
DISSERTATION
Presented in Partial FulfiUment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By Natalia A. Humphreys, M.S. *****
The Ohio State University 1999
Dissertation Commitee: Approved by Professor Bogdan M. Baishanski, Advisor Professor Gerald A. Edgar Advisor Professor Paul G. Nevai Department Of Mathematics ÜMX Number: 9919871
Copyright 1999 by Humphreys, Natalia Alexandra
All rights reserved.
UMI Microform 9919871 Copyright 1999, by UMI Company. All rights reserved.
This microform edition Is protected against unauthorized copying under Title 17, United States Code. m a 300 North Zeeb Road Ann Arbor, MI 48103 © Copyright by
Natalia A. Humphreys 1999 ABSTRACT
If Ç is a non-negative integrable function on R, satisfying some rather general con ditions. then the behavior of the n-fold convolutions ''------V------' n times n —>■ oc is described by the Central Limit Theorem. However, the problem of describ ing the behavior of rises in several contexts when
0, where A ^ 0, ç an integer > 2. and it Wtvs shown that converges in some weak sense to the inverse Fourier transform of exp(AC^)). Recently B. Baishanski has argued that in case Re A = 0, the scaling factor is not the "natural” one. Namely, it is known that in case of probability densities the scaling factor is essentially unique, so the “natural” scaling for in complex valued case should be the same as the essentially unique scaling of He has considered two examples and shown that under the natural scaling one obtains analogues of the Central Limit Theorem of a new kind when Re A = 0. We have obtained more general results of the same nature. Our main theorem is
11 Theorem . Suppose that ip is a complex-valued function such that
p{u) e L^(R) n L"(R) for some s > 1
| 6 L \R ) ? [nip{t) = i ^2 4- G{t), t £ U — a neighborhood of zero , r= p where p, g G N, 2 < p < q, Op, Op+i , . . . , a,, /? are real, > 0, 3 > 0. G 6 C'^iU), G"{t) = 0 { t‘>-^), G{t) = t -)■ 0. Let the scaled n-fold convolutions be given by \/\u)x{c), where ipxic) = cr„>p^'‘^(cr„Ac), a„ = . A = Then \/Xib\ will exhibit one of two types of very regular divergence: I. If p is even, then V X ' i p x i c ) = F(c)ua(c) + ri(c), where F[c) is an integrable function given by F(c) = B\c\~'^ exp { -F lc l''} , (1) |ua(c)| = 1 and Q |rA(c)| < for |c| < In^ A, A > A , where (2) B, E, C, e,7 ,p and A are positive constants that depend on p and q. ui Moreover, the family (ua) satisfies: for every interval [a, 6] not containing zero and every arc I of the unit circle lim —m {c : c € [a. 6],ua(c) € /} =-^{ardength of I) A —t o o b — d 2 7 T 2. If p is odd. then \/Â|V'A(c)| = ^{c)tx{c) + rA(c). where [ 2F(c), i/o o , F{c) being defined by (1),^ < t\{c) < 1 andr\{c) satisfies (2). Moreover, the family (éa) satisfies: for every interval [a. 6j not containing zero and for every subinterval I of [0,1] lim —^ m { c : c G [a,6],ÉA(c) e 1} = - f , ^ — \ —*oo 0 — a JI \/\ — X As an application of our main Theorem we obtain a continuous analogue of Girard’s asymptotic formulae for norms of powers of absolutely convergent Fourier series. IV Dedicated to my family ACKNOWLEDGMENTS First and foremost, I would like to express my deep gratitude to my teacher. Pro fessor Bogdan M. Baishanski, for his helpful discussions and the suggestion of the topic for this dissertation. His intellectual support and warm encouragement were invaluable. His wisdom, patience, humor and kindness were much appreciated. I am especially grateful to him for his careful and detailed criticism of all my written work. I am thankful for his kind permission to use his recent work "Norms of Pow ers and a Central Limit Theorem for Complex-Valued Probabilities” as part of my Introduction. The works of D. M. Girard “A General Asymptotic Formula For Analytic Func tions” and of G. W. Hedstrom “Norms of Powers of Absolutely Convergent Fourier Series” were also of a great help in creating this dissertation. I would like to thank the members of my committee, Professors Gerald Edgar and Paul Nevai. for their time and patience in reading this work. Thanks to the OSU Mathematics Department faculty members who have taught me during the past years, as well as the department itself for academic and financial support. I am especially grateful to Professors Ranko Bojanic and Saleh Tanveer. VI I wish to express my deepest gratitude to my husband, John Humphreys, for his constant support, unbounded love and unwavering faith in me. I would like to thank my dear parents. Alexander and Vera Bondarev for their love, laughter and for giving me perspective throughout my time in graduate school. I am most grateful to all of my relatives and especially to my grandmother, Natalia Slepjan-Bondareva, for their love and for believing in me. I wish to thank my parents-in-law, Charles and Ann Humphreys, for their contin uing emotional support and kindness. Finally. I would like to express my deepest gratitude to my friends: Tona Dickerson and Thomas Stacklin for their vital encouragement. Without all of these people this project would not have been possible. vu VITA July 26, 1972 ...... Born - Leningrad (now St. Petersburg), USSR (now Russia). June. 1993 ...... M.S. Mathematics. Department of Mathematics and Mechanics. St. Petersburg State University, Russia. June. 1993 - .A.ugust. 1994 ...... Graduate School Fellow. Department of Mathematics. The Ohio State University. September. 1994 - present ...... Graduate Teaching .\ssistant. Department of Mathematics. The Ohio State Universitv. PUBLICATIONS Bondareva (Humphreys), N. A. and Bondarev, .A.. S.. (1998) Representations of the Functionals on the Regularly Normalized Operator Ideals, (Russian), Izvestiya GETU. Fundamental mathematical models and their application in electronics and automatics. 512: 13-24. Bondareva (Humphreys), N. A. and Bondarev, A. S., (1996) Lattice Nuclear and Lattice Integral Operators, (Russian), Izvestiya GETU. Collection of scientific works. Mathematics, 450: 10-18. vui Bondareva (Humphreys), N. A. and Bondarev, A. S., (1994) Absolutely Summing Operators, (Russian). Izvestiya GETU. Collection of scientific works. Mathematics. 472: 13-18. FIELDS OF STUDY Major field: Mathematics Specialization: Mathematical Analysis Studies in .-Vpplied Complex Variables Professor Saleh Tanveer .■Approximation Theory Professors Ranko Bojanic and Paul Nevai .A.symptotic Analysis Professor Saleh Tanveer Complex .Analysis Professor Francis Carroll Differential Equations Professor Saleh Tanveer Real Analysis Professors Gerald Edgar and Paul Nevai IX TABLE OF CONTENTS Abstract ii Dedication v Acknowledgments vi V ita viii CHAPTER PAGE 1 Introduction 1 1.1 The Classical Central Limit Theorem ...... 1 1.2 Signed and complex-valued probabihties ...... 2 1.3 Our problem ...... 3 1.4 Two cases ...... 4 1.5 Why the new scaling? ...... 8 1.6 Another kind of central limit theorems ...... 9 1.7 Norms of powers ...... 19 1.8 Thesis structure ...... 21 2 The model problem 24 2.1 Introduction of n-fold convolutions ...... 24 2.2 Scaled n-fold convolutions ...... 25 2.3 Asymptotics of / a (0) and / a (0) 26 2.4 Asymptotic behavior of / a ( c ) for c > 0 ...... 28 2.5 Asymptotic behavior of / a ( c ) for c > 0 ...... 52 2.6 Integral / a ( c ) in case c < 0 ...... 57 2.7 Asymptotic behavior of lACc)! d c ...... 60 2.8 Asymptotic behavior of |A(c)| dc ...... 63 2.9 ZCnorms of n-fold convolutions and ...... 67 X 3 More complicated phase function: polynomial 69 •3.1 Introduction of n-fold convolutions ...... 69 3.2 Scaled n-fold convolutions ...... 70 3.3 Location and representation of the stationary points ...... 73 3.4 .-Vsymptotic behavior of for p - even...... 75 3.5 .A.symptotic behavior of rpxic) for p - o d d ...... 90 3.6 .\symptotics of L^-norm of n-fold convolution ...... 98 3.6.1 Estimates valid for all p ...... 98 3.6.2 Asymptotic formula in case p is e v e n ...... 102 3.6.3 .\symptotic formula in case p is odd ...... 105 4 Asymptotics of the n-fold convolutions in general case 108 4.1 Introduction of n-fold convolutions ...... 108 4.2 Transition from the smooth to the analj'tic function ...... I l l 4.3 .Asymptotics of -norms of n-fold convolutions in the general case . . 130 5 Asymptotic formulae in case p = q 132 5.1 Asymptotics of the scaled n-fold convolutions ...... 133 5.2 .Asymptotics of the norm of n-fold convolution ...... 139 Bibliography 142 XI CHAPTER 1 INTRODUCTION This work represents the search for an analogue of the Central Limit Theorem for complex-valued probabilities. We will explain how the question arose, state the prob lem and the main results, discuss the work of some other authors and point out the differences between their version and our version of the Central Limit Theorem. We will close the Introduction by describing the structure of this work. Several passages in this Introduction have been taken from [1]. 1.1 The Classical Central Limit Theorem Let there be given a sequence of mutually independent identically distributed random variables - having a continuous distribution function. Denote their variance cr^ and mathematical expectation /z. Let us consider another random variable 'In = + ^ 2 + -----1- Central Limit Theorem states that uniformly in [a. 6] as n oo. If yP is the probability density of any variable Çk in the above sequence, then = ipir,p*ip*---*ip is the probability density of the random variable rjn- Thus, the Central Limit Theorem can be reformulated using the language of probabil ity densities. Namely, as n —>• oo, the scaled n-fold convolution ay/nx + n/i) conv-erges weakly (in 6]) to ^ ex p {~ t}- 1.2 Signed and complex-valued probabilities It was discovered in late 1950s that the Central Limit Theorem was also valid for non-positive "probabilities". They have been applied in different areas by a number of authors: .A.. Zhukov [15] in numerical analysis, V. Krylov [14] in partial differential equations, R. Hersh [10] and K. Hochberg [11] in distribution theory-. Namely, if ^ is a complex-valued integrable function, then as n —>• oo its n-fold convolution n times scaled under general conditions in an obvious manner, converges weakly to an ana logue of a Gaussian distribution, which is of the form of inverse Fourier transform of exp{.4x*'} for some constant A ^ 0 (real or complex) and integer q. Recently B. Baishanski called our attention to a different scaling of n-fold convolutions in some special cases. He has shown that under his scaling n-fold convolutions diverge in a very regular oscillatory manner, thus finding a new type of analogues to the Central Limit theorems. In his work Baishanski [1] looks at two typical examples, functions and p„, defined by VeV = exp - ff), po{t) = exp - f '') , (1.1) where g is an even integer, q > 2. He considers n-fold convolutions n times ri times and introduces new scaling (/Pe.n(c) = and z/'o.nCc) = He then shows that under his scaling ■tpe,n{c) and ’0o,nic) diverge, but diverge in a ver\' regular oscillatory manner, which implies, in pgirticular, that \iVe,n{<^)\ converge to the function of the form S exp(-£’|c|‘'), where B, E are positive constants dependent on q. 1.3 Our problem Our problem was to extend results obtained for the special functions Çe and defined in (1.1) to some rather general class of complex-valued functions of a real variable. We described the behavior of = n —>-oo n times for "probability density” complex-valued functions tp such that (/p(i/) € L^(R) n L*(]R) for some s > 1 (1.2) u-p{u) e L'(R) (1.3) In addition, we assume that the Fourier transform (p oï if satisfies: Iv5(^)| < ^(0) = 1 for every t # 0 (1.4) and has the following representation in a neighborhood of zero: In p{t) = + G((), where (l.o) ReAj=Q for j = 2 ,.... Ç — 1; Re ^ 0 : G is twice continuously differentiable in a neighborhood of zero and satisfies G"(t) = as t -)■ 0. We note that the condition (1.5) is automatically fulfilled if is real-analytic at t = 0 and satisfies condition (1.4). There are two parameters that characterize asymptotic behavior of p at the origin: integers p and q. q being defined in (1.3) and p = min{j : j > 2. Aj ^ 0}. Thus, q characterizes |:^| and p characterizes arg(y). Parameters p and q play the principal role in our problem, and there is the crucial difference between cases p = q and p # g- It follows that if the function p satisfies the above conditions, then ReAi = 0. ReAq < 0. 2 < p < Ç and q is even. It also becomes clear that without loss of generality we may assume Ai = 0 and iAp < 0. 1.4 Two cases Thus, we have two cases: case p = q (we call it the "easy case") and case p ^ q (we call it the "delicate case”). For the easy case we consider the scaled n-fold convolution tpnic) = CTn(p^”>(cr„c), where cr„ = n^/^ (1.6) and we prove the following Theorem 1.4.1 If the function (p satisfies conditions (1.2),(1.3),(1.4) and (1.5) and if the parameters p and q are equal, then the scaled n-fold convolutions defined by (1.6). will converge pointwise and in the L^(R)-norm to the inverse Fourier transform of g {j:) = expiAxf) ipn{c) = (^) 4- E n (c), where |i?„(c)| < K min { ^ . for every k < ^. This theorem is an analogue of the Central Limit Theorem for the complex-valued probabilities in case p = q, and the inverse Fourier transform of exp(.4x'') is an analogue of the normal distribution. In the "delicate case” the right scaling is given by \fXW\[c). where Wx[c) = anip^'^^a^Xc), cr„ = A = (1.7) and the following Theorem holds true. Theorem 1.4.2 If the function satisfies conditions (1.2), (1.3), (1.4) and (1.5) and if the parameters p and q are distinct, then the scaled n-fold convolutions \/\Wn, defined in (1.7). will exhibit one of two types of very regular divergence 1. If p is even, then v /  ' 0 a ( c ) = F ( c ) u a ( c ) -f- rA(c), (1.8) where F{c) is an integrable function given by F{c) = B|cr^exp{-E|c|"}, 7 = p = — (1.9) AP - 1) P - 1 |ua(c)| = I and C |rA(c)| < —^ for |c| < In^ A, A > \ . where A |C| B, E. C, e and A are positive constants that depend on p and q. Moreover, the family (ux) satisfies the condition: for every interval [a. 6] not containing zero and every arc I of the unit circle lim —^—m{c : c 6 [a. b], ux(c) E 1} = ^(arclength of /) A —►OO u — Cl 2 7 T 2. If p is odd, then \ Z A | t/;a ( c ) | = $ ( c )« a ( c ) + s a ( c ), ( 1 . 1 0 ) where { 0, if c F(c) being defined by (1.9), 0 < tx{c) < 1 and I«a(c)| < for \c\ < In'^A, A > A for some positive constants e and A. Moreover, the family (éa) satisfies: for every interval [a, 6] not containing zero and for every subinterval I of [0, 1] lim — m{c : c 6 [a,6|,ÉA(c) Q 1} = - f f - - - A-+00 b — a J [ \ / l — A few remarks are in order. 1. Even in the special case olp ^ q when is defined by p(t) = exp{z('' — ( € R (1.11) the proof of the fact that ||rA|lri —>■ 0 as A —>■ oo requires some delicate estimates. In that case we do not need the stationary phase method. This is true even in the simplest case of all: (p{t) = exp(zt^ — [2] when the decomposition (1.8) is obvious. 2. In the special case (1.11), the explicit expressions for ua and tx are ua(c) = exp |-z ^CA|c|?^ - 0 j , tx{c) = cos |C A |c|^^ - ^ j j . where C depends only on p and q. .3. The non-symmetry in case p odd may be surprising. To understand this fact, we look again at the special case (1.11). We fix c < 0. Then the function t.{x) = —rx + x^ is increasing and, taking t as the new integration variable, we obtain that f exp {zA(—cx-f- 1 '') — dx = [ dt. (1.12) J — OO "/ —OO where (p is an analytic function on the real line. Therefore, as |Aj —>■ oc, the integral on the right tends to zero exponentially as the Fourier transform of an analytic function. We obtain then from (1.7) and (1.12) that 0 as A —)• oc for ever}' c < 0. 4. If ^ satisfies conditions (1.2) and (1.3), but condition (1.4) is replaced by a similar one at finitely many points, in the neighborhood of which ^ satisfies (1.5), the behavior of can again be analyzed with the use of Theorems 1.4.1 and 1.4.2. Indeed. Since the behavior of is superposition of contributions from each of these finitely many points, the main task is to describe the behavior of in case there is only one such point. W ithout loss of generality we can assume that point to be the origin, and that ^(0) = 1; in other words, we can assume that (1.4) holds, and then the behavior of is described by Theorem 1.4.1 if p = q, and Theorem 1.4.2 if p # g. 1.5 Why the new scaling? In [i] an argument was made that the scaling (1.7) W^(c) = anip^^^{a„Xc), a„ = A = is the natural scaling in case p ^ q- Here is that argument. It is known (see Theorems 1 and 2 on pages 40-42 of [8]) that if G„ is a sequence of probability distributions with the property: there exists a sequence (cJ„) of positive reals such that Gn{SnU) converges to a proper (1 13) distribution G{u), then the sequence (Sn) is essentially unique. We will define a sequence of probability distributions and will show that it satisfies property (1.13) with = cr„A, but first let us give two definitions. .A probability distribution G is improper if there exists i/q such that G(u) = 0 for !/ < f/Q and G{u) = 1 ior u > i/q; G is proper if it is not improper. Expression “■(()„) is essentially unique" means that if (77^) is another sequence of positive reals with the property that GniVni^) converges to a proper distribution ff((/), then T}n/àn converges to some positive constant k, and H{u) = G{ku). Consider the n-fold convolutions in case p even. Define the probability distribu tions Gn by It follows that Because of (1.7) and since as A —7- oc ~ \/Â||F||^i, we obtain that G„(o-„Ac) -)■ G{c) = T T ^ I F{t)dt IK 111 J-00 The distribution G is proper, and we see from above that (1.13) is satisfied with = (j„A, and, therefore, the scaling (1.7) is essentially unique. 1.6 Another kind of central limit theorems Several authors ([10], [11], [12], [13], [14], [15]) had realized that signed probabili ties make sense analytically, had found a corresponding Central Limit Theorem and used it in the theory of generalized functions, in measure theory, partial differential equations and numerical analysis. (The first seems to have been A. Zhukov [15]). There is a big difference between the work of mentioned authors on one side, and our results on the other side. To illustrate that difference, we present here versions of the Central Limit Theorem, due to Zhukov [15], V. Krylov [13] and K. Hochberg (Theorem 2 in [11]). We change notation in order to make comparison with Theorems 1.4.1 and 1.4.2 easier. We quote a Lemma which appears in the proof of Zhukov's theorem, use a particular case of Krylov’s result and slightly generalize the one of Hochberg. Zhukov’s version We begin with the analogue of the Central Limit Theorem given by A. I. Zhukov [15]. where he investigates the behavior of a difference operator. He argues that the main question of any numerical method - precision question - looks much more compli cated on integration of partial differential equations (p.d.e.) than on integration of ordinary differential equations (o.d.e.) (by integration of a differential equation we mean finding its solution). If in case of an o.d.e. the error of the result is a number, for p.d.e. it is a function. Suppose we integrate an equation of a hyperbolic or parabolic type, with initial conditions given at y = yo, using a difference equation. Let us compare for some ij = yi > yo the solution u* of the difference equation with the solution u of the differential equation. Passing from the function u to u* may be viewed as an action 10 of some operator W u* — W u which depends on y as a parameter. Clearly, the properties of the operator W com pletely determine the error of the numerical process. In his work Zhukov obtains the asymptotic behavior of the operator W as y oo for the simplest case of a linear equation with constant coefficients, given on the whole real line. It turns out that the theorem which holds for this operator is analogous to the lim iting theorem of probability theory. In general, the probability theory language is not applicable since then we would have to operate with such notions as negative probabilities, random variables with zero and negative variance and such. Neverthe less. some methods of the probability theory can be easily generalized on such cases. Zhukov’s work is dedicated to the example of such generalization. To formulate Zhukov’s result, we need to introduce some notation. Consider an equation: du du d^u d^ u — = a „ a + a , — + a , ^ + --+ a„g ^ with constant real coefficients, given on R. Choose some r and set u“(i/) = u(yo, f/), = u(yo + r,u) Passing from -uP to may be viewed as an action of a linear operator F : = Fu°{u) 11 Suppose now that F is approximated by some difference operator G so that u*{u) — Gu^{u) — 4- m/i). m where m belongs to some fixed finite set of integers. VVe will say that operator G has index q . if for any polynomial P{u) of degree less or equal than y—1. GP{u) = FP{u). L'sing Fourier transforms, Zhukov proceeds to show that in the neighborhood of ( = 0 a'-{t) = 0{t)u°{t), where ii'(t) = where the function ip can be written as »•(*) = 53 =Ë^(«)* m fc=0 Here the coefficients ak can be calculated as and thus are the moments of the operator G. Expanding In into Taylor series around i = 0, he finds that ln'0(t) = ^ (1.15) fc=o ■ 12 where 3k &re the cumulants of k-th order (or semi-invariants) of the function i/;. which are connected with the moments ak- Zhukov states that in order for G to have index q it is necessary and sufficient that for all k < q the moments and, therefore, semi-invariants 3k of G and F were the same. In this case it follows from (1.14) and (1.15) that uj{t) '= \aw{t) — In 0(f) = -^ -(-l)’ra, {ity 4- o{t3), t —>■ Q and if we consider functions = ^{u) = )- f e—'m dt. J —oo then we obtain that u*(i) = Taking inverse Fourier transform. Zhukov concludes that the operator W which trans fers solution u^(iy) of the differential equation to solution u’{u) of the difference equa tion and expresses the error of numerical integration u ' = Wu^ may be realized as a convolution of with the function •p: OO p{u — t)u'- (f) dt / ■OO If we now set u(")(t/) = u(yo + "T,u) = F^u°{u) u'(")(t/) =G "u°(t/), 13 then where ip^^\u) = ^ e '"‘/"(t) dt and the following Lemma holds true Lemma 1.6.1 Let G be the operator of index q >2. Let 7 = - (-l)V a ,, cr„ = n?!?!?, ii!n{c) =o-„£p('*^(cr„c). Then as n oo the functions ip„ (c) converge, as generalized functions, to the inverse Fourier transform of the function exp R em arks. 1. Note that in his Lemma Zhukov uses the following scaling c = cr~^u. It is similar to our scaling of (1.6), in case p = q (compare with the result of Theorem 1.4.1). 2. Note also that Zhukov assumes very weak conditions on the function u. That is why he obtains only weak convergence (convergence over a corresponding space of generalized functions). Krylov’s version We now turn our attention to another example, given by V. Krylov [13], who for mulated the analogue of the Central Limit Theorem for the case of a sequence of mutually independent random variables prescribed different non-positive probabil ity densities. For the purposes of our discussion, we formulate a particular case of 14 Krylov’s theorem, namely, his result for the sequence of identically distributed random variables. Theorem 1.6.1 Let ^1 . ^ 2 sfcT--- be a sequence of mutually independent, identically distributed random variables, prescribed, in general, by non-positive probability density satisfying the conditions: 1. For some even q oo roo roc / 2. There exists a constant C such that f u^\(p{u)\ du < C f u‘^ A For an arbitrary r > Ü lim ^ f u'^\ip{u)\ du = 0, n-»oo bfl 71^1 >rB„ where = nb*^, b defined in (1.16). 4. lim m ax-;^ [ u‘‘\(f>(u)\ du = 0 . ^ 0 n>l A 15 Then the probability density of the normalized sum converges weakly over the space Z to the inverse Fourier transform of the function ^(x) =exp{-^}; ^(u) = lim = — / exp < —7- ^ - it'x > dx n-^00 27ry_^ L (q)'- J Rem arks. 1. The precise definition of weak convergence over the space Z is given in [4]. The sequence of functions converges in Z, if the sequence of their inverse Fourier transforms. Xnit), converges in the space K, that is, all these functions, which are real with continuous derivatives of all orders and with bounded support, vanish outside a certain fixed bounded region, the same for all of them, and converge uniformly together with their derivatives of any order. 2. Note that in his Theorem Krylov uses the following scaling 1/ = 6cr„c. It is similar to our scaling of (1.6) in case p = q and to Zhukov’s scaling (Lemma 1.6.1). 3. We now explain the connection of the above result with solution of a partial differential equation. Recall that the fundamental solution of the heat equation is Gauss’ law for any y. Namely, consider the heat conduction equation in an infinite rod du _ dy dv^ 16 with initial condition u(y, 0) = 0 (thus the fundamental solution). If we assume that u(O.O) = uq. then the solution is of the form r _ 2 u{u, y) = Uq < 1 1= / e ^ dx It follows that the solution of the equation dy du't has the form of the limit probability density for arbitrary* y. Hochberg’s version Let us now present a version of the Central Limit Theorem due to Hochberg (Theorem 2 in [11]), which we generalize slightly and, in addition to notation, also change terminology. Theorem 1.6.2 / / € L^(K). 1^1 < 1 and if ^{t) = exp(^t‘^ 4- o((*)), t 0. then n«c^^"*(n?i/) converge, as tempered distributions, to the inverse Fourier transform o/exp(.4f'^). Moreover, if q is even and Re A < 0, then converge weakly on the space of Fourier transforms of integrable functions. Let us observe that |<31 < 1 implies Re A = 0 i{ q is odd, and Re A <0 iï q is even. Proof. Let d be any integrable function, then J du = J d{t)(p" dt ^ (1.17) 17 since the assumption \(p\ < 1 shows that the integrands in the second integral are dominated by the integrable function Since Cp{t) = exp(*4t'^ + as t —)• 0, we obtain that for every t ^ = exp(4t*' 4- o(l)), n -)■ oo It follows then from (1.17) that for any 0 integrable lim [ d[i>) {u If Re.4 < 0, then q is even, so the inverse Fourier transform of exp(4f^) is an integrable function G, and the integral on the right is equal to f d{i/)G{u) du, which proves the last part of the theorem. The remaining case is when A is purely imaginary. In that case we assume that 6 is an arbitrary function from the Schwartz class, and denote again by G the inverse Fourier transform of exp(4f^), which in this case is a tempered distribution. In that case e{t)exp{At^) dt = {G,d) = {G,9) I' so that (1.18) implies that, for any 0 G «S, \im{-^(p^"^\u‘^),d ) = {G,9) n-^oo which ends the proof of the theorem. □ There seems to be only one connection between Theorems 1.4.2 and 1.6.2: obviously, if converges in then it converges to the same limit weakly on the space of Fourier transforms of integrable functions. There are, however, several important differences: 18 1. For distributional convergence it is irrelevant whether |y| attains its absolute maximum at only one or at several points, for L^-convergence that is an essential difference; 2. if p q. then the parameter q is irrelevant for distributional convergence, but it quite relevant for our description, given in Theorem 1.4.2; 3. if p 7^ Ç, then the scaling we use (which was justified earlier) is different from the scaling used for distributional convergence; 4. the distributional convergence approach has been well motivated by applications in numerical analysis, partial differential equations, ... ; 5. that approach is more general; G. there are aspects of the behavior of the n-fold convolutions which are detected by Theorems 1.4.1 and 1.4.2, but not by more general theorems. 1.7 Norms of powers D. Girard [5] considered an analytic function / on the closed unit disk, OO OO f{z) = 11/11 = ^ la^l < oo i/= 0 0 Assuming that l/(z)l < /(I) = 1 for Izl < 1 and ? In / (t) = ia t + i 'y ^ art*" — ydt** -I- G (t), t Ç T, t —>• 0 r= p 19 he obtained the asymptotic formula for j|/”|| as n -> oo. We can reformulate his problem as follows: If a = (a^)o° € describe the behavior of where «bo = a*a*a*---*aasn~^ oo. The characteristic function of the distribution a n times is the function f{z) = . Certain aspects will be simplified if we consider the continuous case instead of discrete: functions y € L^(R) instead of sequences a Çl From Theorems 1.4.1 and 1.4.2, in a very natural way, we obtain a continuous ana logue of Girards Theorem. Corollary. Suppose that if satisfies conditions (1.2),(1.3),(1.4) and (1.5). Then 1. [f the parameters p and q characterizing the behavior of if at the origin are equal, then the -norms of n-fold convolution converge as n oc to the L^-norm of the inverse Fourier transform of the function g(x) = Namely, = ||^~^17||l‘(R) + O (n ”‘) for some e > 0. 2. If the parameters p and q are distinct, then (.4 -I- O for some e > 0. where .4 is a constant dependent on p,q,Ap and ReAq. Let us show how to get the second part of Corollary from Theorem 1.4.2. From (1.7) we have that IM-'IU. (1.19) 20 In ciise p even, it follows immediately from (1.8) and the fact that ||rA||L‘ 0 that ||F||f,i, so that from (1.19) we obtain If p 7^ 9 and p is even, then ||cp‘"^IU^~N/Â|lFlUt, A^oo (1.20) In case p odd. we similarly obtain from (1.10) that + o(l) -7 ^IIFll/^i ho oc = 5-2II^II^7T , ) = 5II^IU‘TV SO that if P 9 and p is odd, then _ l/Âliriki, A-+00 (1.21) Evaluating the integral of F and substituting in (1.20) and (1.21) with A = we derive the second part of the Corollary. 1.8 Thesis structure In Chapter 2 we consider "probability density” functions defined by X{t) = exp(i|i|P - and tpit) = exp{i|t|Psgn(t) - |£|'^}, where the parameters p and q are real and 2 < p < q. We describe the behavior of n-fold convolutions = X * X * X*___*X and = n times (p * tp * 'Ip * - • ■ * Ip a s n —>cx> and find asymptotics of their F'-uorms. n times 21 Note that the "probability density” (p{t) = exp{iC - t ’’}, where p and q are integers. 2 < p < q and q even is a special case of the above two functions for p even and odd. The reason why we can consider the n-fold convolutions with real parameters p and q is that in order to prove the results we do not need aiialyticity of exp{it^ — C'}. In Chapter 3 we consider "probability density” functions defined by 1 \(t) = exp{i ^ Orf - r=p where d. Or € R: p, ^ 6 N, J > 0, 2 < p < q and q is even. We describe the behavior of = asn—>oo and find asymptotics '------V------' n times of their LCnorms. In this case the phase function is of a more complicated nature. It is interesting to note that in case of an even p the asymptotic behavior of the modulus of the n- foki convolutions (and therefore of their ZCnorms) remains the same as in the much simpler case considered in Chapter 2, but the imaginary paxt changes significantly. In this Chapter we use the method of stationary phase. In Chapter 4 we consider 'probability density” functions 22 the results for established in the previous chapter. The estimates of the differ ence are quite delicate and we have obtained them by adapting methods of G. W. Hedstrom, applied in a similar problem [9]. (For that purpose we had to correct an error in an estimate of the difference, and that correction was not a trivial one (the correction is in the proof of Part 3 of Lemma 4.2.4)). Note that unlike which is entire, the function ip is only continuous satisfying some smoothness conditions. In Chapter 5 we consider the case when p = q and the “probability density” functions p satisfy (1.2),(1.3),(1.4) and (1.5). It follows from the definition of the “probability density” function (p{i/) in each Chap ter that its n-fold convolution y(”)(i/) can be expressed as an inverse Fourier transform of <^{t) for all n sufiiciently large, i.e. as an integral. In finding asymptotics of this integral the main difficulty is to obtain its estimation as a function of not only n (which would be a simple exercise), but also as function of the parameter u, as later we integrate in G R to obtain the LCnorm of the n-fold convolution. 23 CHAPTER 2 THE MODEL PROBLEM In the first remark to Theorem 1.4.2 of Introduction we have mentioned that even in the special case of p ^ g when cp(t) = exp{ii^ — for all t € R the proof of the fact that the remainder of the asymptotic formula converges in L^-norm required some delicate estimates. In this chapter we will find asymptotic Ijehavior of the n-fold convolutions of two slightly generalized functions assuming that p and q are real and satisfy 2 < p < q. 2.1 Introduction of n-fold convolutions Define the functions \ xit) = exp{z|t|P - and (2.1) = exp{z|i|Psgn(t) - i € R (2.2) Our goal is to describe the behavior of n-fold convolutions = \* x* ' ' ' * X n times and = -tp * ip * ip if ■ ■ • * tp as n oo. ' S/ n times 24 Since ^ and iv"’ are integrable, the n-fold convolutions can be expressed as their inverse Fourier transform. Thus, for any n and —oc < u < oo we consider = ^ / ° ° dt (2.3) <"l(i/) = ^ / dt (2.4) 2 7 T 7 _ o o 2.2 Scaled n-fold convolutions L'sing (2.1) and (2.3), we write: ^ f exip{—iut + in\t\’’ — dt. 27T 7_oo Denote \ \ 1-2 1 U U A = A„ = n 1 , (Tn = ni, c = - , = - — - (2.o) Making the substitution t = n~^x, we obtain: Y^"^(t/) = — J exp{zA(—cz -t- |z|'’) — |x|'^} dx Similarly, from (2.2) and (2.4), we have (^'^"^(f/) = — [ exp{zA(—cz + |z|^sgn(z)) — |z|''} dx JKCJn 7 —00 Define the scaled n-fold convolutions as >/A/a (c), where /^(c) = cr„x^"^(cr„Ac) (2.6) \/A/a (c), where / a (c) = (J„i/;^”^((T„Ac) (2.7) 25 Thus. A(c) = ^ y* exp{zA(-cx + jx|^) - dx À(c) = :^ [ exp{zA(-cx + |x|^sgn(x)) - |xl’} dx J — CO Studying of the behavior of the n-fold convolutions and as n —»■ oo reduces this way to studying of the integrals I\{c) and I\{c) as X oo. Asymptotic behavior of / a ( 0 ) and / a ( 0 ) differs from that of / a ( c ) , c 0 and I\(c). c # 0 correspondingly. Different scaling is used to study one and the other. We begin with finding the asymptotics of / a ( 0 ) and / a ( 0 ) . Remark. In what follows we assume that C is a positive constant not necessarily the same at each occurrence. 2.3 Asymptotics of /a (0) and / a (0) By definition /A ( U ) = ^ r and / a ( 0 ) = ^ T 2" 7-oc 27T We obtain asymptotics of / a ( 0 ) and / a ( 0 ) as a corollary of the following Lemma 2.3.1 Let p, f/6 R , 2 < p < q and J = Jo Then 26 Proof. Clearly. -X J = / dx -h lim [ {e-^ - dx = .h + lim (2.8) Jo X-*ooJg A'->oo Making the substitution t = Xx^ and applying Cauchy Integral Theorem, we obtain: Ji = I a - p f tp-^e^‘ dt = - e ‘^ A "p r ( (2.9) P Jo P \ P J Consider Jg. .h = \e~^ - l)e'^’’dx = \ g(x)de*^", where g{x) = ------Jof X Jof ipxP-^ Since q > p, we have p(0) = 0 and, integrating by parts, we get: rX XJo = g{X)e^^'’ - / e^^'’g'{x)dx Jo Since g'{x) is absolutely integrable on [0, oo) and g{X) —y 0 as X —> oo. lim J? = O I T I , A — oo (2.10) .V - f oo \X/ Combining equations (2.9) and (2.10) in (2.8), we obtain asymptotics for J: J = i e ‘^ A "p r + O , A -)• oo □ Corollary 1 The scaled n-fold convolutions /a(c), defined by (2.6), will exhibit the following type of asymptotic behavior at the origin as X oo: Aiwo) = ie '# r(i)+ o ( 1 ) 27 P roof. Since —ixj'' + LX\x\^ is an even function, we can write roo -f A(0) = - r -x^+iXx^ =_ -J 7T 7 o ^ Therefore, by Lemma 2.3.1, MO) = ± e'*A -Jr(l)+ o(|) □ Corollary 2 The scaled n-fold convolutions îx{c), defined by (2.7). will exhibit the following type of asymptotic behavior at the origin as X oo: Ai/.(0) = ic o s (|)r(i)+ o (^ P roof. Since sin{A|x|P.sgn(x)} is an odd function, we can write ^ \ { 0 ) = — f G cos Aor^dx = —Re < f ^ dx ^ Jq ^ LVoWo Therefore, by Lemma 2.3.1, ix □ 2.4 Asymptotic behavior of I\{c) for c > 0 In this section we will prove the following 28 Theorem 2.4.1 For any c > 0 the scaled n-fold convolutions, defined by (2.6), mill exhibit the following type of asymptotic behavior as \ oo: \/XIx{c) = F{c)ux{c) + n (c), where F{c) and ux{c) are given by F(c) = -4 pC“ 2(P-n exp{-Bp,,c?^}, Ux(c) = exp |-iCpAc?^ 4- j . Cp = p~^(p — 1) (2.12) and the remainder r\{c) is estimated as follows (2.13) Consider Ix{c): f exp{zA(—car + |z|^) — |z|^} dx —^ J —oo Its phase function is —cx + \x\^. Clearly, its critical point is ^ ' and is movable. To hx it. we make the substitution: z = *’ ‘ t. Then the integral becomes: 2irlx{c) = f exp{ifiT{t)} dt, where (2.14) J —oo a= ~ , T{t) = —pt -\-\t\^ and (2.15) p — 04 A = A X c^, .4 = p~p^ (2.16) 29 Since T'{t) = p(|t|P“ ^sgn(i) — 1), the critical point of the function T{t) is t = 1 and it is unique on R. Let Jx{a) = f exp{î>T(i)} di J — OC Adding and subtracting e"^* /f^exp{i/iT’(t)} dt, we write: Jx{a) = e~^ f exp{iiJ.T{t)} dt — Ri(X.a) (2.17) J — O C where Ri{X,a) = r (e-° - e-^^‘^‘’)exp{i^iT{t)} dt (2.18) J —OC VVe now formulate a lemma that will be used to estimate the remainder i?i(A.a). Lemma 2.4.1 Let g and be real-valued functions and be continuously differen tiable on [a.b].g > 0. Consider rb L = / g{x) ey.-ç{ig.if{x)) dx J a Then (a) If g is a positive, y an increasing and ^ a decreasing function on [a, 6], then 2 g{a) \L\ < {J. ^ { a ) (b) If g is a positive, y? together with is an increasing function on [a, o].. then III a p. (/(6) 30 (c) [f ^ has n critical points in [a, 6 ], then \L\ < • sup 9{t) M a< t< b ^(t) (d) If has n critical points in [a, oo) and linit_*oo = 0, then \L\I r I <^ ------2n + 3 sup 9(1) A* t>o (p'(t) Proof. Integrating by parts, we obtain: L = / g{x) exp{z>(AJ(x)} dx = J a 9(z) exp{i> Estimating in absolute value: 9(b) _ g'(a) ( I , 9(^) f g{x)exp{ip.if{x)} dx < 1 ) 4- dx (2.19) if J a L / ( 6 ) ' ( a ) I Ja I W ( : c ) (a) Since Jr is positive and decreasing, we further obtain: 9(b) _g(a) 2 g{a) iii (b) Since ^ is positive and increasing, 9(b) , 9(a) 2 9(b) + + 1^1 4 V (6) (c) Recall th at if a function h : [a, 6 ] is continuously differentiable and has n extremum points on [a, 6 ], then f |/i'(x)| dx < (2n + 2) sup |h(x)| J a a 31 Hence, from (2.19) 9iff) 9(0.) 9{^) 9{t) < i + + (2 n + 2 ) sup (cl) Since = 0, ^2n + 3 / g{x) exp{imp{x ) } dx < ------sup 9{i) IJ a fJ- t> a □ Note that if we take g{t) = e~“ — and (p{t) =T{t) = |t|P — pt, then we need to consider a function ^o((), defined and studied in the following Lemma. Lemma 2.4.2 Let ( 2 .20 ) ^ g - a P-l ’ t = 1 I Fur any a > 0 there exists a unique point ti E (—oo, —1) such that min II //O < a < 1 — then I. there exists a unique point Si € (1 , oo) such that max ^a(i) = ^ q (s i) l< t< oo 2. 'I'a is increasing on ( — 1 , 1 ) 3. sup |'I'a(t)| < Ca 1 , where C = Cp,, |t|>i 32 4- sup |'î'a(i)| = —^ a e ~ |t| III Ifa>l- 2 ^ . then 1. is decreasing on ( 1 , oo) 2. there exists a unique point Sq € (0 , 1 ) such that max = 'î^q (so ) (—1.1) 2. sup |^a(^)| = —^ a e “ |t|>i P - 1 4- sup < Cq 1 , where C = Cp g l ‘ l < i IV If So is the critical point of'4fa in (0,1), then there exists cvq > 1 such that for all Or > Q o 0 < s q ( q ) < ( 2 . 2 1 ) Proof. Part I. Differentiating for any f, we find that W = e x p (-a K r) . m (2 .2 2 ) 33 where /(i) = - ^ ( l - i i |'" ''s g n ( l ) ) - 2 2 M i { L : 2 1 j _ L l (2.23) p-l (i|9 Let us show that for any q > 0 / is increasing on (— 0 0 , —1). Clearly. 1 + is increasing on (— 0 0 , —1). Developing into Taylor series, we obtciin: it|9 ~ |t|9 ^ s! Since each function in the sum above is decreasing on (— 0 0 , —1), so does Hence. / is increasing on (— 0 0 , —1). Since / ( — I) = > 0 and limt_^_oo/(^) = —0 0 . there exists a unique root of f{t) on (—0 0 , —1 ). By (2.22). this root is also a root of on (—oc ,— 1 ). Note that it is the local minimum of Part II Suppose that 0 < a < 1 — I. Consider (2.23). Clearly. /(I) =0. Differentiating /, we get /'W = i ^ - A W - (2.24) where h{t) = - 1 — sgn(t) exp{a(|tl'^ - l)}(a|t|‘^ - 1) (2.25) 34 On differentiating h we have (2.26 ) The function ^ exp{a(i‘^ — 1 )} is increasing from 1 to oo on [ 1 . oc). On the other hand, since — > 1 , there exists a unique 77 G (l,oc) such that 1 _ Eni -----77’'^'’“ ^ exp{a(77‘^ — 1 )} = 0 a Thus. h'{T]) = 0. Since h'{t) > 0 on (1, 77) and h{l) = 0, /i(t/) > 0. Since h'{t) < 0 on {rj. 0 0 ) and limt-^oo à{t) = —0 0 , there exists a unique ^ 6 (77, oc) such that h{Ç) = 0 . Hence, by (2.24). /'(() = 0. Since f'{t) > 0 on (1.Ç) and /( I ) = 0, /(Ç) > 0. Since f'{t) < 0 on (^, oc) and linif_3c /(() = —oc, there exists a unique si G (Ç, oc) such that /(.si) = 0. Thus, by (2.22),'(r;.(sj=0. 2. If t G [—1 , 0 ], then by (2.23) / > 0. Therefore, by (2.22), > 0. Thus, is increasing on [— 1 , 0 ]. Let t. G (0,1]. By (2.26), if 0 < q < 1 — then h'{t) > 0. Since by (2.25) h{l) = 0, h{t) < 0 on (0,1]. Thus, by (2.24), / is decreasing on (0,1]. Since /(I) = 0, f{t) > 0 on (0,1]. Therefore, by (2.22), > 0 on (0,1]. Thus, is increasing on (0.1]. 3. Let first t > 0. Denote r{t) = and g{a, t) = . Then we can rewrite ^'«(t) as follows: 35 Bv Cauchv’s formula r'K(()) where ^{t) is between 1 and t and the differentiation in the numerator is understood as Let A be either [0,1) or [1. oo). Then g'ict, t) »upi*„(t)|=sup|«^^ < sup ^ sup \H{a. i ) |. (€A r'(t) £€A where we denoted H{a, t) = e x p { -a in > 0 , for ( > 0 r p i Since dH(a. t) aq exp{—ai''} {{q — p-j- I) — aqt‘‘) dt p — 1 H attains its maximum at q-p+l to = (2.27) aq Clearly, io > 1- if 0 < a < 1 — If A = [1, oo), then io is inside A and we have: supt>i |«^a(i)| < supt>i \H{a,t)\ = Hia.to) If i < 0, then j-a _ g-alfC e -“ - < = |^ q (—i)|, where now — t is positive \t\P-^ + 1 36 Thus, the same estimate holds for ( < 0. 4. If _\ = [0.1). then to is outside A and we have: sup l^n(t)| < sup l/f(a. t)| = 1) = 0<£ Part III Suppose that a > 1 — 1. By (2.26). exp{a(t'^ — 1)} is increasing from 1 to oo and < 1. Hence. h'{t) < 0. Since /i(l) = 0 and is decreasing on (1, oo), h[t) < 0 on (1. oc). Hence, by (2.24). J'{t) < 0 on (1, oo). Since /(I ) = 0 and / is decreasing on (1. oc), f{t) < 0 on (1, oc). Thus, by (2,22), ’t^(t) < 0 on (l,oo), i.e. is decreasing. 2. By (2.23). / > 0 for t e (—1,0). Hence, by (2.22), ^^(t) is increasing on ( — 1.0). By (2.26) the function exp{a(t^ — 1)} is increasing from 0 to 1 on [0. l]. On the other hand, since 0 < — < 1, there exists a unique r 6 (0.1) such that 1 _ Ezl T-'z+p ^exp{a(r’ — 1)} = 0 Ot Thus, /i'(r) = 0. r being the point of maximum of h. Since h'[t) < 0 on (r, 1) and h(l) = 0, /i(r) > 0. Since h'{t) > 0 on (0, r) and h(0) = - 1 < 0. there exists a unique Ç € (0, r) such that h(f) = 0. Hence, by formula (2.24) /'( Since !'{t) > 0 on (f. 1) and /(I) = 0, /(€) < 0. Since f'{t) < 0 on (O.f) and /(0+) = oo, there exists a unique Sq G (0, Ç) such that /(sq) = 0. Hence, by formula (2.22) 'I'^(so) = 0, So being the point of maximum of 37 3. If a > 1 — then by (2.27) the point of maximum of H(a,t), to < 1. If A = [l. oo), then to is outside A and we have: sup |'P«(t)| < sup \H{a. t)( = H{a. 1) = ^q(I) = £>l £>1 P — t If t < 0. then e~“ — < —TT—:------— = t), where now — t is positive. l'f»(()l = \t\P-'- - 1 Thus, the same estimate holds for t < 0. 4. If A = [0.1). then to is inside A and, as in (2.28). we have: sup l^a(t)| < sup \H{a,t)\ = H{a,to) < Ca " 0<£ Since So is the point of maximum of 'i'a(t) > 0 on (0,sq(q)). Thus, to show (2.21) it is enough to show that ) < 0 for all a sufficiently large. Let us evaluate it. By (2.22) and (2.23) we obtain: j (p — l)or"':-p+^ exp I — I ) = ------ -2__ + 1 — exp ^ a “ 'j-p+i — a | Lp Since lim Ji— t-p+i — + 1 — exp | a ?-p+: — a | = 1 - < 0 Q—>-00 Lp p - 1 there exists Oq > 0 such that for all o > «o we have ^a(a ) < 0. Therefore, the root So(a) satisfies (2.21). □ 38 Lemma 2.4.3 Let Ri{X,a) be defined in (2.18). I. //O < a < 1 - 2^. then C |-Ri (A,û!)| < Xoi‘1 2. Ifa>l- then Proof. Subdividing the integration range of Ri{X,a) at the critical point of T(i) = —pt + |Z|P and at t = —1, we get Ri{X, a) ~ Ji + J2 + J3 (2.29) where ./i = {e ° - e exp{ipT{t)} dt J-z = f (e~“ - exp{ipT{t)} d J —00 J3 = y (e~“ — exp{ipT{t)} dt 1. Suppose that 0 < q < 1 — By Part 11,1 of Lemma 2.4.2 has a unique point of maximum Si € (l.oo). Thus, by part (d) of Lemma 2.4.1 \Jl\ < — sup |^a(()| = — ^a(Sl) PP t>l PP 39 By Part 11,3 of Lemma 2.4.2 and (2.16) —I C' \Ji\ < - a V = -Q -? (2.30) fj> A Similarly, by Part I of Lemma 2.4.2 has a unique point of minimum ti G (—oc. —I). Thus, by part (d) of Lemma 2.4.1 \’h\ < — sup |^a(()| = — IJ-PtK-l PP By Part II..3 of Lemma 2.4.2 and (2.16) —I C' IJ2 I < (2.31) fj. A By Part 11,2 of Lemma 2.4.2 is positive and increasing on ( — 1.1). Thus, by part (a) of Lemma 2.4.1 I./3 I < — |4r«(l)| < -ae-“ = (2.32) PfM jj. A A Using (2.30), (2.31) and (2.32) in (2.29), we obtain the first part of the Lemma. 2. Suppose that a > 1 — By Part 111,1 of Lemma 2.4.2 is decreasing for t > 1. Therefore, by part (a) of Lemma 2.4.1, (2.20) and (2.16), we find that i-AI < —^a(l) < -ae-“ = (2.33) ' - HP p. A ” > Bv Part I of Lemma 2.4.2 IJ2 I ^ — sup |^a(()| — ~'î'a(ii) p p t< - i P P 40 By Part III.3 of Lemma 2.4.2 and (2.16) IJ2 I < - a e ' “ = (2.34) H A ~ A To estimate 7^. let us fix £■ = ( j) ’ and subdivide J 3 as follows: ./^ = Ki + A'2 4- K3, where Ki = J (e““ - e“"^) exp{i/i(-p£ 1^)}4- dt K-^ = J (e"“ - e”"'*'")exp{ifi{-pt + |£|^)} dt = J {e~° - e " " * ' ' " ) exp{i/i(-p£ 4- | £ | ^ ) } dt By Part III.2 of Lemma 2.4.2 has a unique point of maximum sq € (0.1). By Part IV of Lemma 2.4.2 there exists ûq > 0 such that for all q > ao we have Ü < ■•^o(a) < a t-p^'. Choose ai > oq such that < c. Then V o; > cui it follows that o"''-p+' < 5 and, therefore, 0 < So((%) < q “ TT+t < c For these a ^^(£) < 0 on (e, 1). Hence, by part (a) of Lemma 2.4.1 the following estimate holds: O p-ae'f _ „-Q f' If 1 — < a < oi, then by part (c) of Lemma 2.4.1 the following estimate holds: <1 \Ki\ < — sup|'J'a(i(a))| [6 . 1] 41 Since is a continuous function of o; on a finite interval, Thus, for anv or > 1 — ^ I r , I - C _2. _ E (All < y e -or " (2.35) By Part III.2 of Lemma 2.4.2 'î^q is increasing for i € (—1, so)- Hence, by part (b) of Lemma 2.4.1. the following estimate holds: lA-,1 < (2.36) Integrating by parts twice and using definition of T (2.15), T{t) = \t\^ —pt. we obtain: C i A'sl = | y (e “ - e “‘") exp{zAiT(t)} dt dt By definition of (2.20) we have: 2 - e"“) \^.{e)\ + i^a(-£)| = -4 ------^ < C e - ' Similarly, since by (2.22) and (2.23) Thus. C C c I / __ 1^ | A \ | < r dt Xai\ ^ ' A2af 7_e I l|Z|P-(sgsgn(t) - 1 42 Consider r\( V dt y _ J VKI'’~^sgn(f) - 1 / Let g{t) = e “ - e ""I", ^(i) = \t\^ ^sgn(i) - 1 Then Thus, /2 g" g^" 3 g V , Zgg} + y/ y Clearly. g'[t) — a^|f)''“ Lsgn(i)e~“'‘-‘': g"{t) = a:ç|i|^~"e““'‘^''[ç - 1 — ckç|f|*j /(() = (p - p"(f) = (P - 1)(P - 2)|i|P“^sgn(t) Also, on [—c.cj the function | 1 - < \^(t)\ < 1 + and \^'{t)\ < (p - l)e'’~- If p > 3 or p = 2, then for t € [—e, c] - 3 l 43 We note that the function g"{t) is even. Hence, \g"\ dt — 2 \g"\ dt. Also, g"{t) = 0 for t = 0 and t = ±ia, where (g = (2.37) Clearly, there exists « 2 > 1 - ^ such that V a > 0=2 => t2 < e. Thus, for all a > « 2 we have: r \g"\ d t ^ 2 [ ' \g"\ dt = 2 g"{t) d t - 2 f \ ’\t) dt = ^ ( ( 2 ) - 2g'{s) < J-e Jo Jo Jti For all 1 — < Q < 0 2 we have £ < t2 and thus r \g"\ dt = 2 [ \g"\ dt = 2 [ g"{t) dt = 2g'{e) - 2^(0) = Cae —ae‘> J Jo Jo Thus, we have obtained: p - 1 J \(j"\ dt < C o?, if o > 0 2 > 1 — J \n"\ dt < Coe“ 2 , if 1 — -— - < o < 0 2 To estimate \g'\ dt. we first note that [ Ip'I dt < 2maxl5'(t)| J-e Thus, p - 1 y W\ dt < Co?, if o > 0 2 > 1 — f \ 44 Also. (2.38) Thus, we find that If o > q -2 > 1 ~ then dt < C{af + 1 + e “ ) < C{ctt + 1 ) r i M isgn(i) - 1 If I — < « < « 2 7 then r\( V dt < C{ae-^" + 1 + e -“ ) < C (ae"““ + 1 ). J-, I \|t|P-^sgn(t) - l ) where C is a positive constant that depends only on p, q and c. Since « 2 is finite, choosing it sufficiently large, we get the uniform estimate: r\( i^L(t)i V dt < Cai (2.39) y_, I VI^|P"^sgn ( 0 - 1 / If 2 < p < 3. then ( p - l ) ( p - 2 ) '/( ( ) = |f|3-Psgn(f) and thus we estimate L I (lïf'-'ïïw-l) I * - + '^'1 We estimate the first three integrals just as in case p > 3. Using the estimation (2.38) for Ip I and the condition 0<3—p 45 Thus the estimation (2.39) holds true with a different constant. Hence, using (2.16), we get -£ , 1 _ 2ezlL ie 2 + —a 1 (2.40) Combining (2.33), (2.34), (2.35), (2.36) and (2.40), we obtain that I / \ , C" / _£ -a. 1 _2£=l\ C |i?i(A,q;)| < — I O' ? 4- a «e 2 + —o " ) — '' Thus, the second part of the Lemma is proved. □ We now prove an auxiliary lemma to obtain the asymptotics of the integral exp{z^T(t)} dt in (2.17). Lemma 2.4.4 Consider / = / dt, — 0 0 < a < b < 00 J a Suppose that 1. G"{t) > 0 for all t € (a,6 ); 2 . there exists a unique Iq € (a,6 ) such that G(to) = G'(to) = 0 , G"(to) # 0 and G(t) is four times differentiable in a neighborhood of to Then there exists an absolute constant C = C(G) such that for any p.> Q I 27T e'7 < G''{to) 46 Proof- Subdividing the integration range of I at the critical point t = tg. we obtain I = fi + I2, where rb h f exp{i^G{t)} dt JtQ rto I2 = I exp{ifiG{t)} dt J a We will show that for j = 1,2 / 7T C_ < (2.41) \j2 G " { to /' We will prove this formula for Ii- The argument for I2 is similar. Fix any small J > 0 such that G is four times differentiable and G"{t) 7^ 0 in \t—to\ < S and subdivide the integration range at to + rto-{‘6 pbrà /i = Ai + /1 2 = / +-K / exp{z>G(t)} dt (2.42) ^ (o (0 +^ Let C ' = 2 G " ( t o ) (2.43) Making the substitution v = G{t) in the first integral of the above sum, we have nto+6 G'(t) I J = ^ eM m G (t))^^ dt = I e - . - ï dv 47 where .4 = G(io + <>)• Clearly, f dv = f dv — f dv Jo Jo JA Using Cauchy Integral Theorem, we obtain: r Integrating by parts and estimating in absolute value, we have: 2 C 2 dv / H\/A Ai Thus. \ j - v/^e‘< < E (2.44) I y/J^ /i Consider the second summand in (2.43): /•to+<5 / 1 C * \ K = j ^ exp{,>GW}(^— dt Denote 1 C* 9(f) = 757777 - y e w Then ptQ-^6 K= d{t)exp{inG{t)}G\t) dt JtQ Integrating by parts and estimating in absolute value, we obtain: /"to-hé lim 9{t) + 10 (^0 4- <5)1+/ |9' (f)l dt (2.45) t—*tQ JtQ 48 It is clear that d(t) is continuous on (io,io + ^]- Using the finite Taylor expansion of G ( , - w'+_ ,„)3 + g g p , , _ + O (« - W») G'(t) = G"{to){t — fo) + — H ^ (( — ^oŸ -*■ O {(t ~ to)^) aod the definition of C*. we find that y # ) - C-G'M 1 G'"(to) ' G'wycw 3(G"(t„))3 Thus. 0{t) is also continuous at to. Hence, both limt_,t<, 0(f) and 0(^o + in (2.45) are finite. Similarly, the function 9'{t) is clearly continuous on {to, to + d]. To show that it is also continuous at to, we consider, in addition to the expressions for G and G' above, the finite Taylor expansion of G" G'[t) = G"(^o) + G"'{to){t — to) 4------^ —-(( — fo)^ + G ((t — to)^) Straightforward calculations show that ,. iGlWGW(to) + (G-((o))^ = ------2 ( G n W ------Thus. 9'{t) is also continuous at to. Hence, |0'(t)| dt in (2.45) is finite. Therefore. |A-| < J which, combined with (2.44) in (2.43), implies that < - (2.46) ti 49 Consider -6 112 exp{i^iG{t)} = ** to~^6f Integrating by parts, we obtain: 1 I f G"(t) lAzi < — [1™ G'(t) + G'((o + i) VkWL (G'(i))" . fJ.G'{tQ 4- (f) Since G'{tQ -t- d) > 0, I/12I < — (---I”) fi Using (2.46) and (2.47) in (2.42). we obtain (2.41) for Ii. □ We now obtain estimation of the integral exp{i/iT(t)} dt in (2.17) as a corollary of the preceding Lemma. Lemma 2.4.5 oo P' / exp{iAQ‘»T(t)} dt = —p— ^u\{oi) + iÎ2 (A, a), where oc v X a i ^ ux{a) = exp |-iAa?(p - 1 ) + Proof. Recall that T{t) = \t\^ — pt. Its critical point is t^ = 1 . Let p = Ao? and G{t)=T{t)-T{l) = \t\^-pt+ p-l 50 Then f exp{iXa^T{t)} dt = f exp{ifiG{t)} dt. J —OO J —OO where G satisfies all conditions of the previous Lemma. Therefore, we obtain the result. □ Proof ( of Theorem 2.4.1). By (2.14). (2.17), Lemmas 2.4.5 and 2.4.3 we can write: Ix{c{a)) = ^-^^ua(q) +ai {e~°Rï{\, a) - i 2 i(A,a)). where V A cv 2 Let R\{a) = \ f X o i i (e “Æ2(A,a) - i2i(A,a)) By Lemmas 2.4.3 and 2.4.5. we have: \Rx{a)\ < . 0 < a < 1 - VA \ / (I Çj f 0— I 2p-“2 \ T) — 1 |^A(a)| < - 7 = (e""Q~V + ) , « > 1 - ^----- VA \ / 9 Since < C e"“a “ ^ < C a ~ ^ for 0 < a < 1 — ^ and e~°a~^ < Ca ^ for a > 1 — 2^ . we find that 9 51 / \ -E- Siiice by (2.15) a = f ^ j " '. we obtain: F(c(a)) = Fp—^ = .4pC exp{-5p,,cp-‘ }, O ' ^ where , Bp,, = p' p - l . kA(c(a))| = |FA(a)| < ^ m in Q, and ua(c) = exp | —tCpAc^ + j . where Cp = p~p^ (p — 1 ) Thus, we obtain the assertion of the Theorem. □ 2.5 Asymptotic behavior of I\(c) for c > 0 Consider fA(c) = n J — OC The behavior of ï\{c) is quite different for c > 0 and c < 0. In this section we consider the essential case c > 0 and prove the following Theorem 2.5.1 For any c > 0 the scaled n-fold convolutions, defined in (2.7) exhibit the following type of asymptotic behavior as \ oo: \/Â/a(c) = 2F(c)tx{c) + s a ( c ) , 52 where F{c) is given by (2.11), tx{c) = cos jCpAcS^ - ^ j , Cp = p~A(p - 1 ) (2.48) | s a ( c )| < y= m in ^ (2.49) Since the phase function —cx + |x|^sgn(x) is odd, the integral / a (c) is real: /^(c) = 2 r e-^" cos(A(-cx + x”))dx = 2Re f Jo Jo .\gain. we make a substitution x = f to fix the movable critical point of —cx + xP. Then the integral becomes: I /■°° / a ( c ) = 2 a?Re / exp{zpT’(t)}dt, where Jo C lc W ^ E a = f —J , T{t) = —pt + and p — a'^X The critical points of the function T[t) = —pt + t^ are t = e ^ . k £ Z. Note that only one point, namely ( = 1 . is on the integration path ( 0 , oc). Let oc / e~°‘‘^ exp{ipT{t)} dt (2.50) ■OO Clearly, J a (q :) = e “Re f exp{ipr(t)}dt — Ri(A,a), (2.51) Jo where Ri(A, a) = Re f (e~“ — e exp{ipT{t)}dt (2.52) Jo The following Lemma gives estimates of the remainder Ri(A, a). 53 Lemma 2.5.1 Let Ri{X,a) be defined in (2.52). 1. //O < a < 1 - 2^, then C Ri{X,a) < Aq » 2. If a > \ — ^ , then Ri(X,a) < ja ~ ^ Proof. Subdividing the integration range of Âi(A, o) at the critical point of T(t) = —pt 4- (P. we get ^ i(A,q) = /i+ /2 , (2.53) where h = Re (e~“ — e~°‘^) exp{ipT{t)} dt / 2 = Re f (e"“ - e~°‘‘^)exp{ipT{t)} dt Jo I. Suppose that 0 < q < 1 — Since I\ = ReJi. where Ji was given in (2.29) of Lemma 2.4.3, by (2.30) |/il < A (2.54) Aa-» Estimating I/ 2 I in the same way as | J 3 I, given in (2.29) of Lemma 2.4.3, using (2.30), we obtain 141 < (2.55) Aa« 54 Using (2.54) and (2.55) in (2.53), we obtain the first part of the Lemma. 2. Suppose that q > 1 — Since I\ = ReJi, by (2.33) 2p— t |/i| < — $.(1) < (2.56) /zp A A To estimate I-z- let us fix £ = (j)*' and subdivide I2 as I2 = AA 4- A/2 , where Ml =Re (e~“ — e“"‘’) exp{ifiT{t)} dt pe M- = f {e “ - e “‘’) cos{/xr(t)} dt Jo We estimate A/i in the same way as we estimated Ki in (2.35) and obtain I A/ll < (2.57) Integrating by parts twice, we have for jA/aj: r r^\ / ivir' rni \ ' dt The further estimates are obtained in the same way as for the integral A 3 in the proof of Lemma 2.4.3 and we find that U ( _£ _sl 1 \ , C _z£zi Combining (2.56), (2.57) and (2.58), we find that C ap-t /îi(A,a) < —Q " Thus, the second part of the Lemma follows. □ 55 Lemma 2.5.2 F„ Re / exp{ii.t{-pt + t^)}dt = 4- ^ 2 (^ , 0 ) Jo q; ^ V A where ' C F„p =- \ = COS (A q^(p - 1) - and |^ 2 (A,«)|< ^ yp(p-l) V 4 / Aa" P roof. By Lemma 2.4.5 roo ^ / exp{zp(-p( 4- = —j~zUx{a) + Ë 2 (A, a), 7o Q : ^ V A where C //a(o;) = |-'Aa?(p - 1) 4 - I and |À 2 (A, o)| < \a i Taking real part of both sides, we obtain the assertion of the Lemma. □ P ro o f ( of Theorem 2.5.1). By (2.50), (2.51), Lemmas 2.5.2 and 2.5.1 we can write: / a ( c ( o ) ) = ^ - ^ t x ( û ) + ( e ~ “ ^ 2 (A , a) - R i { X . a)) , where V A 2, ^ ^ — \J-2wp(l - 1) -^'-5 Let Ra(û:) = x/ aï (e~°‘R2{X,a) - Ri{X,aŸj 56 Since the estimations of ^i(A, a) and ^ 2 (A, a) are the same as Ri{X, a) and i?2 (A, a), using the argument of the proof of Theorem 2.4.1, we obtain: < ^ a («) ip-i ^ a~n. a> 1- ^ Since by (2.15) a = , we obtain: - e~” p - 2 i«A(c(a))| = ÀA(a) < ^ m i n Q , i ^ and tx{c) = cos jC p A c ^ ~ J} ’ Q, = p~^(p — 1 ) Thus, we obtain the assertion of the Theorem. □ 2.6 Integral I\{c) in case c < 0 Suppose that c < 0 . Consider again /a(c) = f e-|:^l''e'A(-ci+|i|PsgnW)ja; J — OO In this section we prove the following 57 Proposition 2.6.1 If c < 0, then the scaled n-fold convolution given in (2.7) i s rstiTnated as follow s: Rem ark. A stronger assertion holds true: if c < 0, then as A -> oo the integral Ix(c) tends to zero exponentially. Proof. I. As in case c > 0. we note that 7 a ( c ) = 2 cos(A(-cx + x P ) ) d x = 2Re ^ Jo Jo Consider poo J = g-r"g,A(-ci+zP) ^ Jo Since c < 0 . G'{x) = —c + px’’~^ > |c| > 0 for any x > 0. Integrating by parts and estimating in absolute value, we obtain I I f e~^" Y dx - ÂGÏÔ) Note that the function is decreasing for x > 0. Thus, 1 / e~^'^ V 2 9 Since |ReJ| < |J|, we obtain that IÀWI < ^ (2.59) 58 This estimation will be used for all |c| < 1 (bounded). 2 . Suppose that |c| > 1 . Making a substitution x = ‘ i to fix the movable critical point, we obtain i r I\(c) = 2a'»Re / e exp{inS{t)} dt, where (2.60) S{t)= pt + t^ and p = a t \ Let roo H\{a) = Re / 6'°“^ exp{ipS{t)} dt (2.61) Jo Fix ^ and subdivide Hx(a) as follows H\{a) = Di + Di, where OO / exp{z^(pi + t^)} dt D> = f e~“‘’ cos{p,{pt + t^)} dt Jo Integrating Di by parts, we have: exp{-Qexp{—ae’ gn } I1 r ( e \ dt Note that the function (2.62) 1 + £P- 1 is decreasing for £ > 0. Therefore, |£)^| < ^expl-gg*?} ^ C(£)e “ (2.63) pp(l 4- sP-y Aa? 59 Integrating by parts twice, we have; ID. -K Let (j{t) = e 4>{t) = 1 4- fP L Then ( K it) y ^ ^ _ w 3^ \l4-fP“^/ 0^ The further estimates are obtained in the same way as for the integral K-i in the proof of Lemma 2.4.3 and we find that \n \ ^ ^ -a- , 1 _32:zl\ C ID2I < y I a »e i + - a " I < ya 1 (2.64) Combining (2.63) and (2.64), we obtain C 3P-1 I^A(a)| < yû " L’siug (2.60). we obtain: C 3(p-i) h{c{a)) < —a f - A Since a = j ’’ \ we get: À (c(a)) (2.65) Combining (2.59) and (2.65), we obtain the assertion of the Proposition 2.6.1. □ 2.7 Asymptotic behavior of /_ooI-^a(c)1 dc The purpose of this section is to prove the following 60 Theorem 2.7.1 If the parameters p and q are real and satisfy 2 < p < q, then the L^- norm of the scaled n-fold convolution, defined in (2.6), has the following asymptotic behavior as A —>• oo \/X j 1/a(c)1 dc = Dp^g + O • (inhere P ro o f. Substitution x = —x shows that the function fx{c) is an even function of the param eter c. Therefore, r \h{c)\dc = 2 r \ h i c ) \ d c J —oo J 0 Subdividing the range of integration at we write r | / a ( c ) | dc= f + f | / a ( c ) | dc ( 2 . 6 6 ) 7-00 7|c| |27t/a(c)| = f exp{îA(—car + |r|^) — |x|‘^} dx\ < f dx = - F f . \j— OO I J—OO 9 \ 9 / the first summand in (2 .6 6 ) can be estimated as / |/x(c)| rfc < Ç = O ( i ) Therefore. 61 Bv Theorem 2.4.1 VA I | / a ( c )| dc = 2Ap f c exp{-Bp,,c ? ^ } dc + Rx, where IAaI < |rA(c)| dc. .4, = y 2^ _ p j(p-u and Bp^g = p p-‘ By (2.13) Hence. \/ j | / a ( c ) | dc = 2.4p c exp{-Sp,,c?^} dc + O -OC Clearlv. „-2 _j_ /•°° rx p - j , ^ , J c exp{-Bp,,CP-' } dc = J - y c exp{-Bp.gCP-^ } dc. where r ^ p —2 q /* ^ _ P—2 1 y 1 \ / c 2(p-ii exp{—5p„cp-‘ } dc < / c - \ /  J | / a ( c ) | dc = 2 /Ip y c ■*‘p-‘) exp{-B p,,c^ ) dc + O (2.67) / \ -&T Making the substitution u = ( M \ we have: 2 .4 p [ e x p { - 5 p ,,c ^ } dc = 2 /Ip- --- - p 2(p-‘> f exp{-u} du = Vo ’ Q Jo 2 (p(p - 1 )):' p / 2 . 7T g \ 2 g, 62 Denoting [2{p{p-2 (p (p - 1))5 1))2 _ /f pp \ — V;— -,— and combining the above formula with (2.67), we obtain the assertion of the Theorem. □ 2.8 Asymptotic behavior of |^a ( c )| dc The purpose of this section is to prove the following Theorem 2.8.1 If the parameters p and q are real and satisfy ‘2 < p < q.then the O - norm of the scaled n-fold convolution, defined in (2.7), has thefollowing asymptotic behavior as A —> oc \/X J / a ( c )| dc = Ep^g + O ’ where P roof. Subdividing the integration range at c = ± we obtain: r | / a ( c )| dc= f + ( " ^ r |A(c)| dc ( 2 . 6 8 ) J —oo */—oo ** ^ Since by (2.7) 27t/a(c) = I exp{zA(—cr 4- |x(^sgn(x)) — |x|‘'} dx\ < f dx = -F (-) , IV -oo I J —oo 9 \ 9 / 63 the first summand in (2 .6 8 ) can be estimated as [ IÂ(c)l dc Consider the second summand in (2.68). By Proposition 2.6.1 I |7(c)| dc= / + f \ï\dc< V —oo -/—I •/*—OO (2.70) Consider the third summand in (2.68). By Theorem 2.5.2 \/X f \ïx{c)\dc = 2Ap f c~2(P-i) exp{— cos |CpAc dc 4- - D l +R\, where •OO I I^A / i(p-l) Bp,, =p-.->, Cp=p-^{p-l) By (2.49) /•oo In A A ? 7 a Hence, n/A [ lÂ(t:)l dc = 2.4p f c -‘p-d exp{—Bp ,,c ^ } cos |CpAc?^ - ^ j | dc 4- +0 (^) Denote A' = 2.4p J c exp{—Bp,,c^}lcos jCpAc^ — ^ | | dc 64 Making the substitution A(p — 1) = w, r = j \ we obtain K = J exp jcOS (^UJT — - j (2.72) where 2 ( p - 1 ) 7 T p Denote p(r) = r 2 exp r?^. The zeros of the equation cos [ljt — f ) = 0 in oc) are Tk — — ( ■——r Trfc ) , A: = 0 . 1 .2,... Also, Tt^.^ — Tt = — uJ \ 4 J ^ Subdividing the integration range at we have from (2.72) CÂZ “ I / 7 T \ h=Ep 4-Ep^ / p(r) cos (wr - -) dr " fc=o ' ’■* Clearlv. - _ i . A / [ g{T) icos (w r ~ 7 ) I < f T i dr = —= Jcw-f I \ -4'I Vo \/uJ Hence, K = EpY^^J g{T) cos (ujT - dr + 0 (2.73) By the Generalized Mean Value Theorem there exists € [r*., Vk+i] such that / 7 T \ I 7^+1 / 7T \ J g{r) cos l^wr - - j dr = g{^k) y cos (^wr - - j d r T*+l sin(u;r - f)| = Ui^k) UJ w 65 Therefore, f g{r) CCS (ujT - 0 r f r = -Y^gi^k){Tk+i - Tk) = k=0 ' ’■*= ' ‘ k=0 9 2 /*Tfc+i = - y , 9{r) dr + - T {g{Çk) ~ g(r)) dr (2.74) Consider r *+* R = y igi^k) - g{T)) dr k=0 "/'-k Since g is a regularly varying function, \gi^k) - g(r)| < M\g'{T)\■ \^k - r | < M\g'ir)\ • |r*:+i - Tk\ Hence. f'Ic+l \R\<— y \g'{'r)\dr ^ k=Q Vr.. ’■* Note that g{r) = —r 2 exp < 0 Thus, 1^1 ^ ^ è ^ (E) “p (“ (E) Using (2.74) and (2.75) in (2.73), we get: K = '^ f^ g(T) dr+ o ( ^ ) , u ^ oc 66 Now ^ /-Tfc-Kl /•«> TOO ^ J g{r) dr = g{r) g{r) dr fc=0 Clearly. rrj rx: , c / f7(r) dr < t dr = —j= Jo Jo Using the formula J exp{—'yx°)dx = —7 ~ ? r ^ . (2.76) with J = ^. Q = 7 = we obtain ^ i "p(-") °(;&) = Since £p = we find that A- = £,., + o ( ^ ) , where £ „ = ( j ) W £_illlr (A) Combining the above formula with (2.71). we get r IÀWI dc=E,^+o(^^ Using (2.69) and (2.70) in (2.68), we obtain the assertion of the Theorem. □ 2.9 L^-norms of n-fold convolutions and Theorem 2.9.1 Let ^ and ip be given in (2.1) and (2.2), 2 < p < q. Then as n oc the norms of powers of n-fold convolutions exhibit the following asymptotic behavior: lb‘"'llr' = (^) ’ rri(i-f) + O (In n) ll^Wllr. = ( f ) ’ + 0 (lnn) 67 P roof. 1. By (2.5) ik‘"'iU'=r di^=x r i/a(c)i dc'^à'' J — oo «/ —oo = \/p(p-JOp yx + 0(lnA) \ 7T / rr V zn J Since A = n* ., we obtain 2. By (2.5) = r |TZ;W(f/)l diy = X r | / a ( c ) | ' J — OC • / —oo 2 \ ^ \/p(P Ur x/X + 0 (lnA) .7T/7T / g \ 2 g. Since X = n^~^i. we obtain A") o (In n) = (-YV 7T / q ("\2 g y + 68 CHAPTER 3 MORE COMPLICATED PHASE FUNCTION: POLYNOMIAL 3.1 Introduction of n-fold convolutions Consider complex-valued functions x such that \{t) = expli^Or^'’ - (3.1) r= p where /i. € R i3 > Q, p,q e N, 2 < p < q and q is even. Our goal is to describe the behavior of = x*X*X*'"*X as n —> oc. n times Since ^ is integrable for all n > 0, the n-fold convolution can be expressed as the inverse Fourier transform of Thus, for —oo < z/ < oo we consider = ± (3.2) — 7T J 69 3.2 Scaled n-fold convolutions By (3.1) = — / exp{— 4- i n ^ Ori’’ — Jni''} A -/-30 r=p VV’p [jass from the set of parameters n, u to the set of related parameters A and c. As in Chapter 1, denote A = A„ = (Tn = ni, c = —i-. (3.3) Making the substitution t = n~^x, we obtain: = T "— f +i{-cXx + XupX^+ 9x{x))} dx, LTTtTyi J —oo ? where 6x{x) = A (3.4) r= p + l .Again, as in Chapter 1. we scale the n-fold convolutions and introduce \/X (p x{c), where ^ a (c) = Ac) (3.5) Thus. Ux{c) = ^ J’2 ^exp{-dx'?+i/iA(c, x)} dx, where (3.6) hx{c,x) = —cAx + AopxP + 9x{x) It follows from the definition of 0 a(x), (3.4), that for sufficiently large A it has the following properties: < C X ~ ^ In'' A if |x| < In^ A for some r/ > 0, A: = 1.2,3: (3.7) XxP-^ (AopxP 4 - 0x(x))^*^^, k = 0, 1,2 are monotone in each [—AT^e, 0 ] J (3.8) and [0, A ^ e ] for A sufficiently large and e sufficiently small 70 R em ark . W ithout loss of generality we may assume that Op > 0. Indeed. Let P(t) = apP + ■ If Op < 0, we can write Pit) = —\ap\t^ + Ylr=p+i a.nd then, taking conjugate of %(")((/), we can put it into the same form w ith Op of the opposite sign. In the following we will find asymptotics of ^x{c) for the cases p-even and p-odd as A ^ oo and |c| < In^ A. We will prove the following Theorem 3.2.1 If the function x i-s given by (3.2). then the scaled n-fold convolu tions \/Xivx{c). defined in (3.5), will exhibit one of two types of very regular oscillatory divergence, depending on p be even or odd: I. If p is even, then \/«/;a (c) = F{c)uxic) + rA(c), where F{c) is a function given by F[c) = B\c\~^ exp {-E\c\<^} , 7 = ~ — ^ (3-9) - 1 ) P “ 1 |uA(c)| = 1 for all c and A and kA(c)l < for |c| < In^ A, A > A. where B, E. e. A and C are some positive constants. Moreover, the family (ua ) satisfies the condition: for every interval [a, 6 ] which does not contain zero and every arc I of the unit circle m{c : c G [a, ô],ua(c) G 1} —)■ ^{arclength o f I), A —>• 00 6 -Ü '• ' ' ' 2-k 71 2. If p is odd, then \/X\ipx{c)\ = ^{c)tx{c) + s a ( c ) where { 0 , ifc < 0 2 F(c), i f o O F{c) being given in (3.9), 0 < tx{c) < 1 and 1sa(c)| < for \c\ < A, A > A, A"|c| where e. A and C are some positive constants. Moreover, the family (tx) satisfies the condition: for every interval [a.b\ which does not contain zero and for every subinterval I of [0,1 ] 1 2 f dx m{c : c 6 [a,b],tx{c) 6 /} -)■ — / A —y oc — a IT JI vT r2 R em arks. More precisely. 1 . The coefficients of integrable function F. defined in (3.9) are -S = y / 2 ;r(p^~ 1 ) ^ = f^ipop) 2. The function ux is { exp{i/iA(c,XAc) + if} , if c > 0 exp{ihx{c, ~yxc) + if}, if c < 0 , where hx{c,x) is given by (3.6), xxc is the critical point of hx{c,x) for c > 0, —yxc is the critical point of hx{c,x) for c < 0 . 72 •‘i. In case p odd y/Xrpx{c) = ^(c)t;A(c)fA(c) + Sa(c) and the functions i\ and v\ are h i e ) = cos (/ia(c, tac) - hx{c, -(Txc)) + ^ j T ü a ( c ) = exp [ / i a ( c , txc) + h x [ c , - < 7 a c ) ] | . where txc and —ctac are critical points of hx{c,x) for c > 0 . 3.3 Location and representation of the stationciry points In this section we determine the existence of the stationary points of the phase func tion h x i c . x ) and the interval where they are located. We also find their asymptotic behavior as A —>• oo. We will assume that (c( < In^ A. The following result provides the location of the critical points in different cases. Theorem 3.3.1 Let Qx = In^ A, |c| < In^ A. Then I If p is even, then is monotone increasing on [—QA»f^A| uud has a unique zero in this interval. 1) If c > 0, then that zero is xxc E ( 0 , Q a ) and |h^(c, r)| > Ac for x G [—QatO]. 2) Ifc < Ü, then that zero is —yxc € (—QajO) and |h^(c, z)| > Xc for x G [0. Qa]- II If p is odd, then 73 1) If c > 0, then there exist two zeros r\c OMd —a^c of x) such that txc € (0 . î I a ) O-Tld — (T\c € (—QajO). 2) If c < Q. then the function h\{c,x) does not have any critical points in [-0 ^ .0 ^ ]. h\{c.x) > A|c| forx e Proof. We will prove the case p-even and c> 0. If X 6 [0. Qa]> then using the first property of the function dxi^), (3.7), for = 1 we find that —Ac -t- ^ApapX^"^ < —Ac 4- ApOpX^“ ^ + 6'x{x) = < —Ac 4- 2XpOpX^~^ Show that both bounding polynomials have zeros in (0. Q a)- Let Ti[x) = —Ac 4- ^ApOpX^"^. Then Ti(x) = 0 ** xj = ‘ > 0- Since c < In" A. ^ < iv In^ A. Hence, x% < Qx- Similarly, if we let 72(x) = — Ac-f-2ApapX^“ ‘, then T^(x) = 0 <=> xo = ( 5^ ) ^ Since c < In" A. < A'ln^ A. Hence, x-2 < Qx- Hence. h^(c.x) also has a zero in [O.Ha]- By the second property of the function #A(x). (3.8). this zero is unique. If X E [—Q a .O]. then \h'x{c,x)\ > I - Ac-r ^Apopx^"^! = Ac4- ^ApOplxlP"^ > Ac The other cases are proved similarly. □ In the following Lemma we give the representation of the critical points. 74 Lemma 3.3.1 / / x* is any critical point of h\{c,x) in the interval [— for |f‘! < In^ A and even or odd p, then as X —y oo |x“| = ^ 1 + O ^A~?^(ln A)''j j for some T] > 0 P ro o f. Let us prove it for x\c - the critical point of hx(c, x) for even p and c > Ü. Since x\c < Q\ and Q\ = In^ A, we obtain: h'xic. Xxc) = 0 <=> -Ac 4- XpCLpX^Xc'^ + ^a(^Ac) = 0 By the first property of the function, ^^(x) (3.7), for A; = 1 it follows that -c + uppx^;'(l + O ^A'ï^llnA)")) = 0 Hence. □ 3.4 Asymptotic behavior of -ipxic) for p - even In this section we will prove the first part of Theorem 3.2.1. It will consist of several steps. Suppose that c > 0. 75 ( 1 ) In this part of the proof we find a small interval around the stationary point where the main contribution to the asymptotics comes from and give the estimations of the integral over the intervals outside of this neighborhood. Proposition 3.4.1 Let 7t(A) = | < cr < fix = In^ A, Ü < c < In^A. Consider the interval around the critical point of h\{c,x) forp even and c > 0. Ai = [rxc(l - 7r(A)), rxc(l + 7t(A))] C [0, fix]- Then 2ttipx{c) — / exp {— + z7ix(c,x)} dx 'A, - AA-'c for some constant C > 0 . Since R = [—oo. —fix] U [—fix, tx] U A i U [sx, fix] U [fix, oc] where tx = xxc(l — 7t(A)) and sx = xxc(l + 7t(A)), we can write 2rripx{c) = / exp { -/)x^ + i/ix(c, x)} dx + T + I2 where h = f + [ exp (-/?x^ + i/ix(c, x)} dx (3.10) J — CO J rt\ r^x I2 = I + exp {—/Jx''+ z7ix(c,x)} dx (3.11) J-Qi Va, 76 Lemma 3.4.1 Let I\ be given in (3.10). Then |/il < CX~^ for any /x > 0 Proof. It is easy to show using integration by parts that if L, /? > 0. q > I. then J[l e~’'"^‘' dx < ~ -r— where C = (IV— (3.12) Since h\{c.x) is a real-valued function and q is even, using (3.10) and (3.12), we have |/i| < 2 f e x p {- 0x‘(} dx < -^exp{-dn%} Jiix Hence. \h\ < (Inx)^q-3 {-^(InA)^''} < C X ^ for any /% > 0 □ To estimate exp {—(dx^ 4- ih\{c,x)} dx and exp {— 4- iTiA(c.x)} dx we will use the following J. G. van der Corput's type lemma. Lemma 3.4.2 Let h be a function such that h"{t) is of constant sign on aninterval [u. 6 ] and h'{t) 7^ 0, t € [o, 6 ]. Let j3 be some positive constant. Then ,-b 77 P ro o f. Since h'{t) ^0, t e [a, 6], integration by parts yields I J a I 6 ±q— I +iiq /r e — J l + A 4- ./^ \J a h’{t) \i: We have Jl < 2 max f — 1 “ !«.6] ViA'(*)iy and Jl < '’■Sf (4ïï) * - ' W Now since /i > Ü and h" does not change sign on [a, 6 ], 6 1 1 I dt '(()!" lA (/%'(())" a h'{a) h'{b) and this completes the proof. □ Lemma 3.4.3 Let I-i be given in (3.11). Then P ro o f. Since h\{c,x) is monotone in the interval [—Qaj^a]» h" is of constant sign. Thus, Lemma 3.4.2 is applicable to the intervals [—Ox, and [sa^^a]: where h\{CjX) has no zeroes. Since hxic.x) is increasing on [—Qa,^^a] and its zero x\c G {t\,sx), x)| is decreasing on [-Qa,^a] and increasing on [sa.^a]- 78 Therefore. 1 1 1 1 max and max [n.nxl |/i'a (c, j:)| |/ia (c, sa )| (A)| Calculate |/i';^(c. sa)|- Using the first property of the function dx(^)r (3.7), for k = 2. we find that h\{c, Sx) = h\{c, Sx) - / i ' a ( c , xxc) = [ & % ( c . u) du > KX f du = JXXC JfVr = KXis^-^ ~^xc') = ^ ^ X c ' ((1 +7r(A))P-' - 1) > KXck{\) Therefore, 1 < h'xic, Sx) AT-"c Similarly. 1 C h'xic, tx) Xl-a □ Proof (of the Proposition 3.4.1). By Lemmas 3.4.1 and 3.4.3 since |c| < In” A C 2-n-wxic) - / exp + i/rA(c,x)} dx < j/i + lo] < TYT A i-'c □ (2) In this step of the proof we will use the stationary phase method. It will consist of three estimations. 79 Lemma 3.4.4 ruJi -3x^ +thx[c^) _ gï/ii,(c,x>c)g-/3xlc , e”‘(z i da \L h'l{c, xxc) Jo C for some e > 0 . - A&+'c where = ^hl{c,xxc)x\cT^‘^{\) Remark. More precisely, the bound is In'* A exp Æ’c ^ j . E > 0. Proof. 1. Let us estimate exp {ihx{c,x)} and exp {—3x‘^} on the interval Ai. First show that in the neighborhood of the stationary point hx{c,x) = hxic.xxc) + ^h"{c. xxc){x - xxcf + 0 (InA)^) (3.13) Indeed. By Taylor's formula hxic.xxc) = hx{c,xxc) + ^hl{c,xxc){x - xxcŸ + ^K'(^^0(x - XxcV for some Ç between x and xxc- Here h"'(c.() = Xa^pip -l){p- 2)e -^ + e'l'iO By the first property of the function 0a(x), (3.7), for A: = 3 we have for x G Ai I -3 a -(x - XxcY 80 Bv Lemma 3.3.1 ^\c = f e ) ’" (l+o(A-*(hAr)) Since |c| < In^ A, = O ^(In and thus formula (3.13) follows. Now. for X 6 Ai -!3x‘' = -^ x \^ + O ((In A) A A -') (3.14) Combining (3.13) and (3.14) and using 0 < 3 cr — 1 < cr for | < cr < ^, we can write: -I3x‘' 4- ihx{c,x) = = -3x1^ + ihx{c,xxc) + ii/i"(c,XAc)(x - xac)^ + 0 (InA)^) Thus. / exp {—dx''+ z7ia (c, (x)} dx = _ g./u(c.iAc)g-^i'L J exp |i^/i';((c,XAc)(a: - xac)^| ( 1 + G’ (x)) dx. where G"(x) =o(A^-^ (In A )i^ ) (3.15) 2. Making a substitution u = |/i"(c, XAc)(a: — xxc)^ in the integral above, we obtain: where 1 Wl = -h%(c, %Ac):CAc^ (^) 81 Therefore. f dx = / _ _ ^ -----_ f ‘ e‘“u-5 du + iTi. 7ai \ h"{c,xxc) Jo where Ki = e:AA(cA\c)g-a%L J exp |i^h"(c, xac)(x - xac)^| G'(r) dz 3. Let us estimate AT(. Clearly, for all A sufficiently large |A:i| < /" |G *(x)|dz JAi Let Then Ac = 4 ( 1 + 0 ^A"ï^(lnA)")y = xl + 0 (InA )A ""^ (3.16) Since x \ c = O (In^ A) and |A i| = 2xAc7r(A) = 0 (A “‘^ln^A), using (3.15) and the assumption cr > |, we have: \Ki\ < J |G*(x)| dx < (l + C A '^ (In A)^^") A"' In^ A • A^-^ (In A) A < CA^"‘*‘^ln® A < e~^^°A-^-' Note that = exp ^ —E c ^ I , where E = 0{pOp)~^. 82 Hence. lA'il < CA'-‘‘‘"ln®Aexp{-£7cA} Since for all c > 0 exp|— < —, the result of the Lemma follows. □ Lemma 3.4.5 Q < — ;------fo T some e > 0 h'^{c,xxc) \/X Rem ark. More precisely, the bound is CF(c)A“^“?^ In'* A. for some t; > 0. where F { c ) i s the function defined in (3.9). P roof. We have h " x { c - X \ c ) y Ap(p - l)opzg where /v, = e - 4 '% — t ______—2 h ' x i c . x x c ) y X p { p - l)up%o Let us estimate \Ko\- < IKzl < -2 \ p [ p - l)OpZg h'x{c, xxc) C I) jxpjp- l)QpXg _ Y /i"(c,rAc) 83 Using (3.16) we have exp{-0{xl^ - xg)} = I + O (A <) Since h'lic.xxc) = Ap(p - l)opX^c (l + O (A ‘)) = Ap(p - l)opxg (l4- O (A ')) we liave HSF— "''-'I Therefore. = O (A -) Thus. \K2\ < CA-"'xo ' On the other hand, since Xq = ( ^ ) » we obtain ; rz =■ Dc exp } "= \/ 2 ^ f (c), V p(p - l)apxS-" I J where 1 D = Y -(pop) and E = 0{poj,) p-‘ Since for any |c| < In^ A, we have [Kg] < and the Lemma follows. □ 84 Lemma 3.4.6 tl/i e‘“u du — y/ïre^* < e > 0 1/ \tQ2lp-l) Proof. Since i: e'“u ^ du = y/ïre^ *, we have rwi ^ / du = y/ire'* + Kg, Jü where nwi roa rco /v3 = / e‘“u“5 du — j e‘“u “ 2 du = — I du V 0 V 0 Vuti Integrating by parts and estimating, we have: /•oo 1 1 roo / e‘“u-5 du < - + - u Ju.’[ u; - " •/u»! IV i Since ^\c and 1 — ^p(p ’ 85 we obtain “■ - Ï {Ï) "(p - (^) ' ' Thus. I^ 3 |< Wi and we obtain the assertion of the Lemma. □ (3a) It follows from Lemmas 3.4.4. 3.4.5, 3.4.6 and Proposition 3.4.1 that y/X-tpxic) = F(c)uu(c) 4-ru(c) where F(c) is an integrabie function given by (3.9); uu(c) = exp [ihx{c,x\c) + if}, where x\c is the critical point of h\{c,x) for p even, c > 0 and IruWI < ^ for some e > 0 and some positive constant C. (3b) We now give an interpretation of the function uu(c). Let us show that for every interval [a, 6 ] which does not contain zero and ever}' arc I of the unit circle m{c : c e [u, 5], Uia(c) € i} ^{arclength of /), A - 4 - oc b - a ^ ^ ' 27T It follows from (3.6) and Lemma 3.3.1 that hx{c,xxc) = A ^-^pcA + ^ 86 where W rite . 7 T ihx{c,xxc) + i - = zAA(c), w here / a ( c ) = - A p C ^ + 4 - ^ Let [a. 6] be any interval not containing zero. To show that the function f\(c) is decreasing on [n,6], we differentiate h\{c,xxc) with respect to c to find: d dhx{c,x) dhx{c,x) dx Ac dx dc r=XXc Since xxc is the critical point of h>(c, x), the second summand in the expression above becomes zero and we have d ^ , dhx{c,x) — “ Axac E (—AflA,0 ). X—XXc which shows that /a{c) is decreasing. Also, from definition Uix(c) = e‘‘^A(c) Let Si,x = {c:c e [a, 6], uu(c) € /} ; f = [yi, <^2 ] C [0.27r) Let xi,x-2 ,... ,Xm denote all integer multiples of ^ lying in the range of /a, i.e. in [/a(M ,/a(û)], an d let x o = f x ( à ) , X m + i = f x ( a ) . D enote 4 = 87 k = O.l m. For l Si,\ n /fc = |c : Xfc + ^ < fx(c) < Xfc + and so /i {Si,x n Ik) = {^k + y ) - /a ^ (xk + y ) = = for some 6 ft- This is true for 1 < ^ < m — 1, but not necessarily for A: = 0 or k = m. However, for A: = 0 we have (•S'/,A Fl Iq) < —------0, A —> GO Similarly, for any choice of Ço G Iq {fx^ï (^o)(^i — %o) -> 0 as A ^ oc. Analogously, if A: = m. then At (5j,a n Ijn) ^ ^ ^ 0, A —> GO and ~ V^2 ( e-i\' (/a ) (^m)(^m+l ^m) ^ 0 fot any G 2tt Thus, we obtain m m /t(5/.A) = (5j.a n Ik) = 2-k^ (6)(zt+i - Zk) + o(l). A -)■ GO k=Q t=0 So the result will follow if we show that m 53 ) (^*)(^k+i — ^k) —)■ a — b, A —>• G O (3.17) t=o Let us observe that if % G Ik are chosen so that f;;^{xk+i) - fx\^k) = ~^k), 88 then m 5 3 ink){:i^k+i - ^ k ) = f^^{Xm+i) - fx\^o) = a -h (3.18) k= 0 Saw if we show that ^ (/a”^) (^) is uniformly bounded by M in A and x £ I. for any I - Unite closed interval not containing 0, then we will have for A: = 0,.... m + I (%) - { / a" ' ) ' % )| < M\rtk - &l < M y . (.3.19) it will follow that the difference of the left-hand sides in (3.17) and (3.18) tends to zero as A —> cx:. which will prove (3.17). Let us show (3.19). Since we have ' (/I(A-‘W))” SO it is sufficient to show that /"(c) is bounded, and /^(c) is bounded away from zero for c € [a.6 ]. Since /^(c) = —xxc < — K c ^ < —/vi < 0 for c € [a,6 ], f'x is bounded away from zero. On the other hand, /"(c) = --^xxc- Since xxc is the zero of h x { x , c) = —Ac + \üppxP ~ ^ 4- 0^^)' we can say that xxc is the inverse function of the function c = c{x) = appx^~^ -h = Qa(^), a polynomial. Since is bounded away from zero for c € [a, 6 ], we get that ^xxc is uniformly bounded for c 6 [a, 6 )- □ 89 (4) Since 1 r'^ — ^ expi-f^f’ 4- i{—cXx 4- Xopx’’ + 0 a { ^ ) ) } dx, J —oo we obtain by changing the integration variable x —> —x that iù\{—c) — iü\{c), where is given by the same formula as i)\{c) above, except that 9x{x) is replaced by 6x{x) = 9x{—x). The case c < 0 is reduced that way to the already examined case c > 0 . 3.5 Asymptotic behavior of iIjx{c) for p - odd In this section we will prove the second part of the Theorem 3.2.1. It turns out that if p is odd, then the contribution to the main asymptotics term of iPx{c) comes only from the case of positive c. If c is negative, then the integral iDxic) is neghgible with respect to the main asymptotic term. This is not surprising since as we established in Theorem 3.3.1 the phase function hx{c,x) has no critical points if p is odd and c < 0. We start with studying the case c > 0. ( 1 ) In this part of the proof we find small intervals around the stationary points where the main contribution to the asymptotics comes from and give the estimations of the integrals over the intervals outside of these neighborhoods. Proposition 3.5.1 Let 7t(A) = A”"”, | < cr < |. Consider the following two intervals around the critical points of h\{c,x) A s = [ta c (1 - 7r(A)),TAc(l 4- 7r(A))] C [ 0 ,Q a ] A 4 = [— 90 Then C 2~ii)\{c) — / - I exp{—3x‘’ + ihx{c,x)} dx < Va , Va 4 Since R = -l.( U A4 U (—0 0 , —Qa] U [0, Qa] \ A 3 U [—QajO] \ A4 U [r^A, 3 c). we can write 2~ii’x(c) = + exp {—âx’’ ihx(c,x)} dz 4 - V; + VA;, Va4 where rOO + / exp { -/iJ z ''+ i/iA(c,z)} dx •oc J Qx h = + exp { - Vz''+ i/iA(c,z)} dx. V[o,nvi\A3 V(-nx,o]\A4 V(-nx,o]\A4'[o,nvi\A3 Estimating I.i as Ii in Lemma 3.4.1 and I4 as A in Lemma 3.4.3. we obtain the assertion of the Proposition. □ (2) In this step of the proof we use the stationary phase method to find asymptotics of ^A, Jsa {-Vr* + z7ia(c.z)} dz as a ->• 0 0 . It will consist of three estimations. Lemma 3.5.1 rW3 / -axi+ihxic.!) I _ ihxic.Txc) / ______? ------/ e‘“u ~ 3 du < Jo for some e > 0 Az+ and f e-d^+'Ax(c^) dx - g^AK-'Ac)g-a<. / [ ' e - “u-3 rf-«< 7a4'A 4 y V -h'lic,-axc) A Jo C < —;----- /or 5ome e > 0 “ A&+'c where 1 W4 = -^h'lic. -o-Ac)o-|c7r^(A) Proof. The proof of the first inequality is similar to the one of Lemma 3.4.4. For the second inequality we note, that making a substitution and then using conjugation we can write r ’'■(A)) / e-Jx’+.A,(c,x) dx= ^ ^ J^ A 4 "/ ~<^\c(.^'rir(X))—o'Ac(l-r)''(A)) ro’.\c(l+>r(A))'•o’.\c(l+>r(A)) / exp {-dx^i — Lhx{c,—x)} dx. where since p is odd, the function —hx{c, —x) = -Xcx + XopX^ - 9\{—x) is of the same form as hx{c,x). Thus, following the argument of Lemma 3.4.4, we obtain the second assertion of the Lemma. □ 92 Lemma 3.5.2 fo r some e > 0 h"),{c,Txc) \A " Ai+'c and y/^F{c) /o r some e > 0 — h 'x {c , —(Jac) \/A Aî‘*’*c’ Proof. The proof is similar to the one of Lemma 3.4.5 □ Lemma 3.5.3 rW3 rW4 ^hx(c,rxc) / e‘“n ~5 du + e *“u 5 cf'u — Jo Jo C —2\/TTVx{c)ix{c)\ < ---- p for some e > 0 , where ux{c) =exp|^[/iA(c,rAc) + hx{c, -o-ac)]| &(c) = cos ( / i a ( c . txc) - hx{c. -(Txc)) + P roof. Let rW3 ft A/ = / e‘“u -î rfu + Jo Jo .Adding and subtracting gthx(c,Tsc) f du = 4 ^ n d Jo ,iAx(c,- 93 we obtain M = v/^exp{i/iA(c. Txc) + «7 } + x/^exp{i/iA(c, - 7 « 7 3 7 Let us calculate the sum of the exponents in (3.20). We have exp |z£a(c, txc) + j + exp |i/iA(c, -ctac) - Factoring out exp j ^ £ a (c . Txc) + ^/iA(c. -o-A c)| , we obtain 7*A(c,r\c) — trxc)+if g — 5 / 1 a ( c ,TAc ) + 4 / ‘ a ( c .— ffA c )— ‘ f ^ . • exp{^/% A(c, TXc) + ^ h x i c , - C T a c ) } = 2 COS ( £ a ( c , Txc) - hxic, -CTxc)) + The remainder |£ 3| is estimated by /•OO I r / e‘“u~3 du + \ If e ‘“n 2 du 7 1773 I 7 1wU4 Arguing similarly to the proof in Theorem 3.4.6, we obtain that I/. e‘“u 2 du I/. e *“ u 2 dn < CA'2+‘^c"^i^ 94 Thus, we obtain the assertion of the Lemma. □ (3a) It follows from Lemmas 3.5.1. 3.5.2. 3.5.3 and Proposition 3.5.1 that \/A|va(c)| = 2F{c)tx{c) + sa(c), where = |Î^a(c)|. tx is given in the above Lemma, F{c) is an integrabie function given by (3.9) and Q | s a ( c ) | < — for some e > 0 (3b) VVe now give an interpretation of the function t\{c). Let us show that for every interval [a, b] which does not contain zero and for every subinterval / of [ 0, 1] r— : c € [a,6],tA(c) € / } - > - [ ./= : as A oc b — a J [ y l — X It follows from (3.6) and Lemma 3.3.1 that {b.x{c. Txc) — hx{c. —CTxc)) = A f —.4pCP-‘ -I----- where ^ (p o p )-A , 0x(c) = O (A '-^ (ln A )" ) . Write ^ (àx(c,rxc) - àx(c, - 95 To show that the function g\{c) is decreasing on every interval not containing zero, we differentiate h\{c, txc) — h.\{c, —a\c) with respect to c to find: d h x { c ,x ) dr:Ac ^ {hx{c, Txc) - hx{c, -axc)) = X—T^c dx ^=-r\c d c dhx(c,x) d h x { c ,x ) dcTXc Oc dx x=- Since Txc and —axc are the critical points of hx{c, x ) , two summands in the expression above becomes zero and we have -J- (^x(c- TXc) — hx{c, —CTxc)) = —Xtxc ~ A(7Ac € ( — 0), dc which shows that gx{c) is decreasing. Also, tx{c) = | cos{A^a(c)}|. Let / = [A, B\ be any subinterval of [0, 1]. Clearly, m {c : c e [a.6] : |cos{AgA(c)}| E /} = m {c : c E [o, 6] : E l{x)} , where /(x) = {e“ : A < I cos(x)| < B} — union of four arcs of the unit circle. By Theorem 3.2.1, in particular the measure interpretation of the function ux{c). lim m {c:ce [a, 6] : £ /(z)} = _ ,) = A->oc ZTT 2 2 r 1 = — (arccosA — arccosB) {b — a) = —{b — a) J ^ dx □ (4) Suppose that c < 0. 96 Lemma 3.5.4 Ifp is odd and c < 0, then C A|c| Proof. To estimate iUx{c) we split the integration interval into three parts: —fix rflx r o o / + + / exp {—l3x‘^ + ihx{c,x)} dx = h + where ■OO J—d\ —\i\ rcc / 4- / exp {— -I-i/iA(c,x)} dx •oo J fix rf^x h = exp {—dx'^ ihx{c,x)} dx J-flx Estimating as /i in Lemma 3.4.1. we have 1/ 51 < CX~^ for any p, > Q. To estimate /g we use van der Corput’s type Lemma 3.4.2. Note that by Theorem 3.3.1 h\{c.x) > —Ac > 0 in [—0^, Since /t"(c, x) > 0 on (O.^a] and A%(c, x) < 0 on [—ilA.O), we can apply Lemma 3.4.2 on each interval separately. Since h\ is increasing on ( 0. Oa]^ 1 1 1 ““ M i” |/iUc,0)l ” A|c| Similarly, since is increasing on [—Q a , 0), we have max 1 1 1 [-»A.o) |/iAl |/Ia(c, 0)| A|c| Thus, the Lemma follows. □ 97 3.6 Asymptotics of L^-norm of n-fold convolution In this section we will focus on finding the asymptotic behavior for the L^-norm of n-fold convolution defined and studied in five previous sections. We will prove the following Theorem 3.6.1 If the function x is given by (3.1), then (.4 + O (n~‘)) for some e > 0, where J = 0, if p is even; 5 = 1 , if p is odd 3.6.1 Estimates valid for all p Recall that \{t) = exp{z OrC — Jt''} and consider the n-fold convolution ^ dt Introduce the following sets of i/: = (t/: M >6n'-^ln2n = D„}, 6= ffn = {u : \u\ < alnn = Bn}, a = 1 ---- Mn = ■ Bn< \v\ < Dn) 98 It follows from the definition of x that for any t/ G R < r dt < C n -î (3.21) J — OO VVe will use this estimation for u 6 VV%. To obtain another estimation of that we will use for v € we will need the simplified version of Baishanski’s lemma. L em m a 3.6.1 Let iu{t,y) = Reln(%(( + iy)) and ly( < ^ . Then = 0 (n"î), n-^oo J — OO P roof. Since w(t, y) = Re I i ^ ar{t + iyY - d{t + iyf L r=P is a polynomial in t and y, we have 9—1 t^(t,y) = ^ C m (y )r- m=0 where Cm{y) are polynomials in y. Since 'ip{t,0) = —0 t ‘^, Cm(0) = 0 for m = 0 q - I- -Moreover, if m = 0,... ,p - 1, then y = 0 is the zero of Cm{y) of order p — rn. Thus, for all bounded y , ifm = 0 , l , . . . , p - l \Cm{y)\ < K2\y\, if m = p,... ,y - 1 99 Thus. ïi y — n we have /•T c / - o o f p - i ') / dt< exp ^ V + T Cmn}-%tr ~ 3nt^ > dt lm = 0 m = p J .Making the substitution un~? = t. we further obtain: n - i n exp I ^ - JuÀ du d —'XI L 171=0 m = p J Now. in order for the powers of n to be non-positive we need that 1 — d{p — m ) < 0 for m = 0 p — 1 and n I ~ 6 ------< 0 for m = p q «/ 1- — The first condition can be written as 0 > m = 0,...,p — 1. The smallest 0 satisfying it is 0 = 1 — In this case the second condition above is fulfilled automatically. Hence. I dt < n 'I I exp i ^rnW\'^ ~ = O (n " ^ . n — : d-'x Lm=o J □ Lemma 3.6.2 For any i/ Ç R there exists a positive constant C such that < Cn~^ exp I [I^} 100 Proof. Suppose that i/ > 0. Since \ is analytic, by Cauchy’s Integral Theorem we can write: r dt= [ r ( z ) e - " " dz, (3.22) ■J—oo “ Prit^ where = (z E C : z = t i, t € K} n « Let tj = Then we have [ (fz = - %y) A J Pnu J —<30 Estimating in absolute value and using Lemma 3.6.1, we get I f dz < e~‘^ f exp{nReln(%(( —zy))} d (< \J p n u J - o o If z/ < 0, then let Pnu = {zEC: z = t + —z, i € R} .\rguing similarly, we obtain the same formulae with \i/\ instead of z/. □ VVe now use the two estimates just obtained to prove Lemma 3.6.3 [ |%("^(z/)| dz/ = f du + O (n q Inrz^ J — OO J t'SAdn 101 Proof. Since R = ^„ U Ai» U 7V„, we write: f + f + f \x‘"Hdl, "/ —OO */ *' vV tn J rfVn Since = a in n , a > 0, using (3.21) f du < Cn~^ f du < Cn~^ Inn J j ^ ' n J o Since D„ = bn^~^ In^n, 6 > 0, by Lemma 3.6.2 f rfi/< Cn~î f exp/ -irrr f du < Cn^~^ exp {—6 In^ n} < J ^ n J o n I n ‘” 1 J < Cn~'i □ Using (3.5), we note that the set jM„ now becomes c € {c : .Aa < |c| < Bx\ , where A\ = , B\ = In^ A A'+ «f—P Thus, by Lemma 3.6.3 J" | x ' " V ) l (/^ = AjT ' |^;,(c)| + \tùx{-c)\ dc + o [ \ - ^ ^ In a ) (3.23) .\t this point the arguments for p even and odd differ. We start with 3.6.2 Asymptotic formula in case p is even By Theorem 3.2.1 for even p, we get \ [ \/X{\iJx{c)\ + \tp\{-c)\) d c -2 f F{c)dc < C f |rA(c)| dc < \J Ai. d Ai d Ai rBx I 102 Thus, we obtain \/A f \tp\{c) + il)x{—c)\ dc = 2 f F{c) dc + O (X~^) (3.24) Jax Jax for some e > 0. Lemma 3.6.4 for some e > 0. Proof. Let us first show that -B 2 J F{c) dc = Fp J X i(p-i) exp | | dx-\-0 [X *) (3.25) for some e > 0. where Fp = ^ x(p-V) ) ' .Making the substitution % = we obtain: j-Bx rBxipaj.) • r _2_1 , 2 / F{c) dc = Fp X exp < —i3xp-^ ^ dx Jax JAxipap)-'’ Clearly. rBxipap)r-5x(pap)-‘ r r , a -1 , /*°° r ° ° rrAx(pap)- / X -(p-u exp ^ —Dxp-i ^ ax = / — / JAx(pap)-^ *■ J 7o 7o — / exp j — j Since .Ax = A '/ - P fAxipap) ‘ r Mx(pop) p_j I X z(p-i) exp < —/3xp-i f dx < I X -(p-^> rfx = -4A(pap) ' / In A \ 2 /•“ f _jz_i exp{-/3(lnA)F^| / X exp < \ dx < C------^ ------JBxipn^)-^ ’■ ■’ (lnA)2(p-‘i Since rAxipap)-^ roc / In A \ 5(^) I + I X exp I — > dx < C I , ) Vo JBxipOp)-'^ ^ ^ \ A "ï-P / formula (3.25) follows. Let us now evaluate the integral in (3.25). Making the substitution u = 3 x ^ . we obtain: J x~-'p-‘> exp | —J x ^ I dx = - — - J du = = EZL q Combining (3.25) and (3.26) we get the result of the Lemma. □ P ro o f (of Theorem 3.6.1 for p even). By (3.23) i-Sx = J ly^")(f/)| du = A J |î/'a(c)1 + |'^a(—c)l dc-l-O^A ■/-plnA^ = -oo J Ax 2n/A j " F(c) dc + O \/Â(A4-0 (A"')) = -t- O (n~')) for some other e > 0 , where □ Remark. More precise calculations show that e = | — cr, |<(T<^. 104 3.6.3 Asymptotic formula in case p is o d d By Theorem 3.6.1 for odd p we obtain as in (3.24) \/X J |c;a(c)| dc = 2 J F { c ) cos (/ia(c, tac) - hx{c, -axc)) + dc 4-0 (A-‘) . e > 0 (3.27) VVe have shown earlier in the proof of the measure interpretation of the function tx{c) (part (3b) for p odd) that i (/ia (c. Txc) - hx{c, -Gxc)) + ^ = AÿA(c), where gx[c) = -A p C ^ + 4- Thus rBx rBxrB, \/ / | |tyA(c)lw a ( c )1 dcdc == 22 / / F(c) |cos {A^ a (c)}1 dc 4 -0 (A~') , e > 0 Jax Jax and the following Lemma takes place. Lemma 3.6.5 Let F{c) be an integrabie function on R. Then lim fr F(c) [cos {A^a (c )}| dc = — f F{c) dc -^-*•0=A—foc 7r Jq Proof. VVe prove that for every interval [a, 6] not containing zero rb 2 lim / |cos {A^a (c)}| d c = —{b — a) A - ^ o o y ^ 7T Note that it follows immediately from this result that the same holds for intervals containing zero. 105 Let fi\ be the function defined on [0, l] by pLx{t) = m {c : c € [a.6 ] : |cos > i} Then by Fubbini's Theorem J icos{AgA(c)}| dc = J ^\{t) dt (3.28) Since I cos u| > f is equivalent to e‘“ € li{t)u{—li{t)), where li{t) = {e“ : cos(s) > t}. we get. using Theorem 3.2.1, in particular the measure interpretation of the function iix(r). that lim m (c : c 6 [a, 6] : € li{t) U = ~ ^^’'^3 th{l\{t)) A—K30 27T Since this convergence is bounded for t € [0,1], we obtain lim f fix{t) dt = [ — a) dt = —{b — a) f arccos(i) d£ = \-^oc J q Vo ^ ^ Jo 2 = —{b — a) 7T So from (3.28) we get the result. It follows immediately that if 0 is a characteristic function of the interval [a, 6], then roo .y rco / 0 (c) |cos{AgA(c)}i dc -)■ - / Therefore, the result is true for the step functions and since the set of step functions is dense in the space of integrabie functions the Theorem follows. □ Remark. A sharper estimation holds true: I 9 r°o \ F{c) \cos {Xgx{c)}\ d c - - F{c) dc U'Aax x ^ Jo 106 "'here r/ = P ro o f ( of Theorem 3.6.1 for p odd). By (3.23) rBx = J"" = \ilj,{c)\ + \M-c)\ dc + o[x-^ p In a ) Lem^3.5.4 ^ dc + f " ~ dc + O (A "Î^ In a ) = JAx JAx ^ ^ 2 \/ J F{c) |cos {A^a(c)}| dc + 0 = Lem^3.6.5 2 \/ ^2 ^ ' F ( c ) dc + o(l)^ Lem^3.6.4 + o ( l ) ) = ^ = ^ (yl + o(l)) where R em ark . More precise calculations show that ||9<"'|U. = ( 4 + O (A -)) . where 107 CHAPTER 4 ASYMPTOTICS OF THE iV-FOLD CONVOLUTIONS IN GENERAL CASE 4.1 Introduction of n-fold convolutions Consider a complex-valued function ç such that Ç E L^(R) n £■*(]&) for some s > 1 (4.1) E L'(R) (4.2) Suppose that its Fourier transform = f du J — OO satisfies \(p{t)\ < y(0) = 1 for every t # 0 (4.3) and has the following representation in the neighborhood of zero 1 9 ${t) = exp(iat 4- i ^ Orf — + G(t)} = exp{^^ ■+■ G(t)}, (4.4) r=p j=l 108 where G is twice continuously differentiable in the neighborhood of i = 0 and satisfies G"{t) = ast-^Q. We assume that fj,Or e R; p,q € N. It follows from conditions (4.1) through (4.4) that a is real, J > 0, 2 < p < q, q is even, G'{t) = O(t^) and G(t) = 0(1^+^) as t —y 0 . We note that the condition (4.4) is automatically fulfilled if (p is real-analytic at i = 0 and satisfies condition (4.3). Parameters p and q characterize asymptotic behavior of ip at the origin, q being defined as Re Aj = Q for j = 1, 2 ,..., g — 1; ReAq^O and p = min{j : j >2. Aj # 0 }. Thus, q characterizes \ip\ and p characterizes arg(^). Our goal is to describe the behavior of 93^"^ = ip*ip*ip*---*tp as n —y 0 0 . ■ V " n times Note that condition (4.2) insures that ^ and are bounded. The following properties of ip follow from the definition: Lemma 4.1.1 lim ^{t) = 0 £—►±00 V Cq > 0 3 0 < 5 < 1 : |i| > Co \ip{t)\ < I - d' (4.5) There exists an G N snch that for all n > N the function is integrabie. Proof. (1 ) The first property follows from Riemann-Lebesgue lemma, since 93 6 L^(R). 109 (2 ) Since i = 0 is the only maximum of ^ on R, iimt_>±oo = 0 and since ^ is a continuous function, the second property follows. (3a) Suppose that (4.1) holds with s = 2. By Parceval’s identity [ \0 {t)\'^dt= f \^p{u)\^ du < oc J _.x; J —oo Since by (4.3) < 1 , for any n > 2 /'“ i5(!)r*= r dt < r\$w\ut< oc J —J — OO J — OO Thus, if (/J G L^. then is integrabie for all n > 2. (3b) Suppose that (4.1) holds with s > 2. Then since tp Ç. L^f\L^ implies p G we are in case (3a). (3c) Suppose that 1 < s < 2. By generalized Plancherel’s Theorem if G I*, then p G Z,*’. where s' is conjugate exponent. Thus, p E L^'. 2 < s' < oc. Hence, since < 1 . for all n sufficiently large (n > s') •oc r O O rC O / i^(()r = / i^(f)r-''. iy(()K < / i^(()r'< oc -oo — oc J — oo Thus. is integrabie for all n > s'. □ Remark. Without loss of generality we may assume a = 0 in (4.4). Otherwise, consider p^"\u — an) instead of p^"''{u). It follows from the third property of the above Lemma that the n-fold convolution p*p*p*..-*:p can be expressed as the inverse Fourier transform of Thus, for large n and — oc < j/ < oc we consider ^ I " dt (4.6) 110 4.2 Transition from the smooth to the analytic function To a function -ç satisfying conditions (4.1) through (4.4) (with a = 0 in (4.4)) we correspond the function x defined by ? x{t) = exp{i ^ Orf - t e R r=p The function \ is of the special type considered in the previous Chapter (see (3.1)). We have described the behavior of = Theorem 3.2.1. n times Denote g{t) x(^) a.nd fit) ^(t). The estimations obtained in this section will lead us to the conclusion that in order to obtain the asymptotic behavior of and its I'-norm it is enough to find asymptotics of and its L^norm. Namely, we will show that ,^W(f,) = xW(t,) + A^(W, where R„ satisfies the following Theorem 4.2.1 Let R,i[u) = — x^"^(f/), tn = Then there exist positive constants Cq and p. such that I If\u\ > pn, then |E»(f/)| < II //Con'i+i < |(u| < p and p is even or odd with w > Q, then 111 I l l IfO < |w| < Cqti ?+' andp is even or odd with w > 0, then there exists a positive constant K such that \RnH\ < Cn~^-^^\w\-^+Cn^-f‘\w \f^e x p {-K n \w \^^ for some p > 0 : IV For p odd, w <0 c I&.M I < n\w\ V For any tw 6 R lAnMI < C n - Proof. Part I. Let us show that there exist positive constants Cq and p such that for \u\ > pn Cn /: dt Choose Co > 0 so that the logarithm (p(t) = ln/(t) is continuous on [—cor^o]- Let us split R into three intervals: (—oo, — c q ) U [—eo, fo] U ( c q , o o ). Suppose first that t G [—eo-eo]- Then we can write eo dt exp{n (f){t) — iut} dt / 60 Integrating by parts and estimating in absolute value, we obtain: «o tVt (0 / dt - dt 60 —60 112 Since !/(()! < 1 for all i € K, the above integral is bounded by r. def 1 r\( ! V dt = var[_,.,,.| — /-«O I - t y / n0 '{t) — iu Let us require that f def \u\ > 2n - sup l^'l = 2n ■ sup = fin (4.7) [—«o.ïo] [—eo,«o] f We wull show that for all u that satisfy (4.7) 1 ^ Cn var[_eo,eo] ; 'nd/(t) — iu ~ u^ Indeed. Taking the derivative, we have: I 1 a ( ^ ) V - L {n0'{t)—iuy\ dt The last integral is bounded by £ ♦ ^ ,1-1 Since o' is of bounded variation, using (4.7) we see that the last expression is bounded by Cn Cn — var[_*„,,o]0 < which proves our claim. Thus, we have shown that if i G [—6 0 , eo), then -eo r ( ( ) e - ^ Co Cn (4.8) — Co - 1^ Suppose now that t e (eo, 0 0 ) or (—0 0 , —eo). 113 Integrating by parts twice, we obtain: n r ° ° r dt = -—nt)e-“^ + - / r-'{t)f{t)e-^''^dt = eo Ao n 4 - -IT 11/ r - x ( ) r ( ( ) e - r Thus. / " r ( A - — 11/ J U) eo Note that by (4.5) |/(i)l < 1 — for |(| > cq- Hence, the integrand ^f')' can be estimated as: Kr"7')'! = (n -i)|/”"Y'| + |r"7"|< < C{n - 1)(1 - . |y ,|2 + ( ! _ ^)a-,V_l|y:V| . |y„| Since /' and /" are bounded and |/^ | is integrable, we obtain: H n < ^ ( 1 _ (4.9) J to — lU 60 Similarly. —60 (4.10) — lU Clearly, oo reo r 114 —60 to the first integral in (4.11); . r(t)e -' to the second one and to the - i f / 60 third one to obtain: I/: IJ^ I/* —to 4- n .\ovv each term can be estimated using (4.7) as 1 1 n\0 '{eQ)\ r(eo)e-‘"'“ T 1- — < ir(fo )i < n©'(eo) — iV w + iuri0'{€o)\ "|.^'((o)| (tT) 2n|y(eo)| < ir(6o)l < ir(6o: 1/2 - \u(f/{eo)\n\ Simiiarlv. tl/€Q 1 1 4- — n Therefore, we finally obtain It remains to note that the same argument holds for g. Thus, there exist positive constants such that for \u\ > /in which concludes the proof of the first part of the Theorem. □ 115 Parts II through I\' Clearly. Let Splitting the integration range at ±a:„ and estimating in absolute value, we have: - x'">(<^)l < I r ( /”(*) - 9’'{t))e-'‘ dt \J -Ctn + [ /" (i)e -“^ dt + / g’"{t)e-"'^ dt (4.12) 'U|>Qn I i‘/|f|>Qa The last two summands in (4.12) can be estimated as follows Lemma 4.2.1 For any f satisfying (4-3) and (4-4) 1/ < Cexp{— }. (4.13) where 7 and C are some positive constants. Proof. Choose any cq > 0 such that for |t| < eo / has representation (4.4) and |G(£)! < $ . Since for all n sufihciently large [-Q „,a„] C [—eo,eo], we can write —iut r /"(£)e-‘"‘ dt < \ r f-{t)e-^''utw\r f-{ty dt (4.14) \J Cin \Jan I I •'Co 116 Using (4.4). we estimate the first integral above as: \ f rfij < [ exp {—n0t‘’ n\G{t)\} dt \Jan \ Jan Since |G(t)| < it follows that + n|G(t)| < Thus, the above is bounded bv: / I = eo exp I | and we have: r Pity < eoexp Jan Since there exists N such that Vn > N is integrable on R. using (4.5), we estimate the second integral in (4.14) as: I r /"(t)e-"" dt < (1 - 6 )^-^ r\f^(t)\ dt < G(1 - Sr = Cexp{-na}. t o J to where cr = In > 0. Combining the above two estimations, we obtain: / r{t)i < Gexp{—7 n<>+‘} \J a„ Similarly, I r-ctn / n ty dt < C exp{— } J —OC □ Remark. Since both / and g satisfy conditions of the Lemma above. 2-k\R,,{u)\ < I r ( / " ( ( ) dt + Gexp{—7 n«+‘ } (4.15) J — 117 The rest of the proof is dedicated to the estimation of the first summand in (4.12) r (/"(()-g"(t))e-"'dt J — O n for different values of u In the interval [ [—Qr„,a„| we have: J = J exp I —itwn + in ^ Orf j e — l] dt. where we denoted w = — (4-16) n Let S{t) = -wt + /h,(() = - 1] (4.17) r = p Then £»n hn{t) exp{m5(t)} dt /•O n In the following Lemma we determine the location of the critical points of S'(t). t € {—Ctni On). Lemma 4.2.2 Suppose that t G (—«n^ctn)- Then there exists a constant Ci < pa^ such that 1) If p is even, then for any w S' is monotone increasing in (—o;„,q„). If |u;| < C i n ~ ^ , then it has a unique zero to 6 (0, a„) for positive w and to G (—q„,0) for negative w. 118 2) If p is odd and Q < w < C'ln” -'-»-*, then S{t) has two critical points, toi ond fog, such that fot 6 (—a„.0) and fog E (0.a„). JJ If p IS odd and w < 0, then S(f) does not have critical points in (—a„,û!„). P roof. Let P(f) = Opf + ^ sufficiently large P PL P" are monotone on each [—On, 0], [0, o„] and in each of them < |P'(f)| < Co|f|^“L where Ci < pOp and Co > pap Suppose that p is even and 0 < u; < C\n~. Then for f € (O.On) -w + Cit^~^ < —w +papt^~^ + ^ rarf~'' < —w + Cot^~^ r=p+l Since 0 < u; < C in“ ‘»+‘. each bounding polynomial has a zero in (0. o„): -w 4 - C t P =0<^=>f=(—J € (0,a„) Similarly. f w \ ^ Cl -w + CofP ^ = 0 4* f = f — j e (0, Q„) since — < 1 Thus, 5'(f) also has a zero in (0 ,O !„ ). Since P' is monotone there, this zero is unique. If f € (-a „ ,0 ) then 9 |S'(f)| = I - w + popf"-' + Y1 rOrC-M > I - W + CifP-'l = w + Ci\tr^ > W r= p + l Other cases are proved similarly. □ The following Lemma gives representation of any critical point of the function S that is in the interval (-Q „,a„). 119 Lemma 4.2.3 Let |w| < Cin ^ and îq be any critical point of S in (—a„ ,0 !n). Then |fo| = (1 4- f ) , where e = O ^ , u; 0 Proof. We will consider the case w > 0. Let Iq be a critical point of S in (-a „ ,a „ ). Clearly, rortl'^ =pOpfg"^ |^14- ^ r=p+l \ r=p+l / Since to < n ~ ^ , we have ~ O Therefore. 5'(éq) = 0 <=> —w+paptQ~^ |l + ^ = 0 V r= p + l P “ p / and so 0 r= p + l P “ P □ In the following four lemmas we give the estimate of hn(t) exp{inS(t)} dt for p even and odd and w positive and negative. Lemma 4.2.4 Let p be even, w > 0 . Then there exist constants Cq > pOp and C > 0 such that 120 _ p—• 1 (IJ If w > C qti i+* then C / hn{t) exp{inS{t)} dt < \Jo nw (2) For any w > 0 hn{t) exp{m 5(t)} dt nw \IV f—a , (2) If 0 < w < CqU ^ , then j y /i„(Z) exp{m5(Z)} rfi < C n ^+ C n^ exp | —A'nu;?^ | for some K > 0 and 0 < p < 1. Proof. (1) We first show that there exists a constant C q such that if (i- > CqU then 15'(t)l > ^ (4.18) Indeed. On [0. tv„] f ^ < n ^ , r = p,p + I, • • ■. Hence, < Ct^-^ < C n~^ r = p _g%l r —1 Choose C q = 2 C . Then, if w > C qh ■'+*, H U p ^rrt < y, which implies that + > — \S'{t)\ = — W ar Vf ^ W — — 9 r=p r=p Let rctn Bn{w) = / exp{in5(t)} di (4.19) Jo 121 Integrating by parts in (4.19), we obtain: ( I ) exp{mS(()} dt Since G{t) = as t —)■ 0, A„(0) = 0. Hence, estimating in absolute value, we have: ^n(Q^n) + ivarp,.,.., But 1 var[o,a„] < sup I Ay, I va r^ 4-sup vaihr. Therefore. 1 1 1 1 + - sup |/i„|var— 4- - sup var/ir, (4.20) n o n S' where supremum and variation are taken over the interval [0, Q;„]. Let us estimate var^ and sup |^ |. By definition of the variation of a function, /•“" 1 S" 1 1 /•“'* S" 1 /■“" I / 1 \ ' dt < sup dt < ^ ~ L 1(5')^ S' I p— I p—i 1 < sup Ct^~ dt < C sup — ni+^n «+* < C sup — Thus, it follows from (4.18) that 1 sup (4.21) To estimate hn{t) we notice that if t G [0, n“5^], then |G(OI < < Cn~^ 122 Hence. \^nG{t) _ l| = < - 1| < Cn\G{t)\ < Thus. \h^{t)\ = - l]| < < C , 0 < t < a ^ (4.22) Let us estimate var/in- By definition rctn var[o,a„]/in = / |/iU ()l Jo Since - I) + n C we estimate each summand in absolute value to obtain: rotn ran var[o.a„]An < Cliqn / df + Cn / < 7o Vo < Cn (ft + C7n5^ if : " dt < C Thus. '^3^[0,an]^n ^ C" (4.23) Using (4.21), (4.22) and (4.23) in (4.20), we obtain the estimate: |B .(“ )I < ^ (2) Suppose that m > 0. 123 Note that since w > Q there are no critical points of S on [—«n, 0] and we can estimate: —w —pap\ty‘ ^4- ^2 ^ > W r= p + l Integrating by parts and estimating as in the proof of Part (1). we obtain I r° C / exp{m5(f)} dt < \J —Ctn nw (3) Suppose that 0 < zn < CqTI . Let us write pT—S pT+6 pQn Bn{w) = / + + hn{t) exp{inS{t)} dt, (4.24) Jq j t—6 j T-k'é where 6 = n~‘’tQ and r > 0 is chosen as follows: a) If fo = ( ^ ) O < a„, then r = to- This will always be the case if 0 < w < C i n ~ ^ (see part 1 of Lemma 4.2.2). b) If to > then r = a„ and the third integral and part of the second are omitted. This might be the case if C in ~ ^ < (n < C o n ~ ^ . Suppose that r = to. Integrating by parts and estimating, we obtain similar to (4.20) •f<) — d 1 hji(to - 6) I / h„(t) exp{m5(t)} dt + - sup |h „ |v ar^ + I do n S'(to - 6) n 1 H sup vaxhji, (4.25) n where supremum and variation axe taken over the interval [0, to — d']. Since sup[o,to_d] |hn(t)| < sup[o,a„] |/in(^)l and var[o,to- sup |h„(t)| < C and var[o,to_j]|hn(t)| < C (4.26) [0 ,£o— 124 We have on [0. io — •to—i I ÇH dt Since S"{t) > 0 for p > 2, Iq—S r-to—S ^If 1 (4.27) (5')2 S‘ since |5'| is decreasing on [0,to — <)]. To estimate |5'(io — ^)| we use Lagrange formula. We have \S'{to - J)| = |5 '(io ) - S'{to - (f)| = ^ • |5"(Ç)|, where to - 6 < ^ < to Since S"{t) is an increasing function, S"{() > S" (I) > cc" > Cwf^ Hence. \S'{to -6)1 > C S w ^ (4.28) It now follows from (4.27) and (4.28) that var[o,to-rf]^ < C 5 ~ ^ w ~ ^ ^ Combining the above inequality with (4.26) in (4.25), we obtain: '£q —<5 I hji{t) exp{inS{t)} dt < Cn-^5-^w~^ (4.29) .■\nalogously, integrating by parts and estimating, we obtain similar to (4.20) / • a n hn{to + 6) 1 / hn{t) exp{inS{t)} dt < i It + \J t+S n S'{to + 6) n 1 1 1 H— sup /i„ var— + - sup vaxhn, (4.30) n S' n 125 where supremum and variation are taken over the interval [to + Qin]- Since \h^{t)\ < sup^o,^^; and var[t„+i,a„]l/in(t)| < var[o,a„i(/in(t)(, by (4.22) and (4.23) sup |/in(i)| < C and var[fo+i,Q„]|/i„(t)| < C (4.31) [iO+<î.“fll As for the interval [0, to — d>], we have on [to + à', a:„]: ^ jy i 2 |S'(t„+,S)| Now by Lagrange formula we have; 5"(to -I- ti) = 5"(to + d) — 5'(to) = 5S"{^), where to + Ç < «n Since 5" is an increasing function, S ' \ 0 > S"[k) > Ctg-" > C 5 w ^ ^ Hence. < sup ■J [t0T-4,Q„! P I Combining the above inequality with (4.31) in (4.30), we obtain: I /■“" / h„(t) exp{m5(t)} dt < Cn~^5~^w~f=^ (4.32) \J'r+rft+S Using (4.22), we find that the second integral in (4.24) is bounded by: rC(jT 126 for some positive constant K. Using (4.29), (4.32) and (4.33) in (4.24), we have: \ B n { w ) \ < C n ^ 0 U ü p - i + CSnwp-'^ exp{— Knwj>-^ \ < < 4- Cn^~‘’w ^ exp{—K n w ^ } The next three lemmas are proved similarly. Lemma 4.2.5 Let p be even, w < 0. Then there exist constants Cq > pOp and C > 0 such that (I) If w < —Co n '9+', then I /■" C / hn{t) exp{inS{t)} dt M —aa n\w\ (2) For any w < 0 I C / /i„(t) exp{mS(i)} dt \Jo n\w\ (2) If —CqU < w < 0, then I r“ I / hn{t) exp{m5(t)} dt Lemma 4.2.6 Let p be odd, w > 0. Then there exist constants Cq > pap and C such that (1) If w > Con~^ then Otn C hn{t) exp{inS{t)} dt nw /■Qn 127 (2) If 0 < w < CqU , then r " I / exp{in5(i)} dt < C n '’u ; ^ exp{— IJ - a „ where A > 0 and 0 < p < 1. Lemma 4.2.7 Let p be odd, w < 0. Then (1) If / hn{t)exp{inS{t)} dt \J —Ctn (2) If\w\ > then I 1°" c / hn{t)exp{inS(t)} dt < \ j - a n n\w\ Parts II through IV of the Theorem now follow from Lemma 4.2.1 and the corre sponding parts of the last four lemmas. Part V The following estimation holds for any it; € R regardless of p being even or odd. By (4.22) I hn{t)exp{inS{t)} dt < C f \hn{t)\dt < Cn f dt < Cn J-nn Jo Jo □ 128 Remark. Let us consider the scaled n-fold convolutions ofip^^\ defined as in Section 3.2 by \/Â^,\(c), where ^ a(c) = and A, = An , = n l-E ^, cr„ = n», i c = We can write If we recalculate estimations for Rn{i') from the previous Theorem for the new argu ment. we obtain that Theorem 4.2.2 There exist positive constants Cq and p. such that I- U |c| > pXi-p, then Ac) < . 1 for every 0 < e < ^ A 5-«|c|l+‘ 2 . If C q < |c| < p \ i - p , t h e n '/XanRn{(Tn>^c) < :i. If Q <\c\ < C q, then \/Âcr„/l„(cr„Ac) for some e > 0 A'|c| providing that p < ^ (p is from part III of Theorem 4-2.1). The above Theorem shows that \/Âa„i2„(cr„Ac) —>• 0 as A —> oo for each fixed c. Thus, Theorem 3.2.1 which describes the asymptotic behavior of the scaled n-fold convolutions '/ ài^x remains valid for '/X'^x. 129 4.3 Asymptotics of L^-norms of n-fold convolutions in the general case In this section we will prove that the Theorem 3.6.1 also remains valid for that is we will show that Rn{u) —>• 0 in L‘(R). Namely, we will prove the following Theorem 4.3.1 Suppose that the function cp satisfies conditions H-l), (4-2), (4-^) and ( 4-4)- U the parameters p and q characterizing the behavior of ip at the origin are distinct, then \p( t)(.4 + 0 ( n *)) for some e > 0. where 6 = 0, if p is even ; 5 = 1, if p is odd. Remark. More precise calculations show that . r i 1 1 3 1 This Theorem follows from the following Lemma and Theorem 3.6.1. Lemma 4.3.1 For p^’^^ and defined in (4-6) and (3.2) the following holds true: oo roo / 1 P I \ du = du + 0 f Innj , n —*■ oo / ■OO J —0 0 130 Namely, if Rniu) = then there exist positive constants C and fj. such that [ \Rn{u)\ d u < C f \Rn{i>)\ du ■= O Inn^ , n oc J - u n ^ P ro o f. We will prove the case p even. Other cases are proved similarly. 1) By Part I of Theorem 4.2.1 f |En(Wl dv /in rn 2 CM 1 \Rn{o)\ do < Cn / n ~ ^ dw 4-C / — dw + / ./in ~ Jo rC on 2 pC on ^ +Cn'^ I —dw + Cn^~^ I , u;p-‘ exp{—/Cni£;p-‘ V die Making a substitution u = Knwp-^ . we estimate the last summand and thus obtain -/XU By choosing p so that p = — p, we find that p = ■ Hence. /*“ " o - „ - l / -pin |Æn(f/)| df/ < Cn 2, Inn J -xxu □ Theorem 4.3.1 now follows from Theorem 3.6.1. 131 CHAPTER 5 ASYMPTOTIC FORMULAE IN CASE P = Q U p = q, the asymptotic formulae become considerably simpler. Suppose that the function satisfies conditions (4.1), (4.2), (4.3) and (4.4). The last requirement in c ase p = q takes the form p{t) = exp{io:t 4- At’ + G(t)}, t -4 0, A = -i3 + m. Let us describe the behavior of =■ qj * ip * N. so that for all sufficiently large n the n-fold convolutions p * p * p * ■■■ * p can be expressed as the inverse Fourier transform of Condition (4.2) insures that p' and p " are bounded. Thus, for large n and —oo y"";") = ^ 7_oo r dt Rem ark. As in the previous Chapter we may assume that without loss of generality a = 0. Otherwise we consider p^^^{u — na) instead of p^^\u). Let and consider the scaled n-fold convolutions 0n(c) = cTn¥’^"Ho’nc) = J x)e~*‘^ dx (5.1) _ I We obtain them by first making a substitution f = n «x and then denoting c = un ?. 132 5.1 Asymptotics of the scaled n-fold convolutions In this section we prove the following Theorem 5.1.1 If the function ^ satisfies (4-1), (4-^), (4-3) and (4-4), and is normalized so that Q = 0. then the scaled n-fold convolutions ihn, defined by (5.1), will converge pointwise to the inverse Fourier transform of g{x) = exp{.4z^}. Namely, 1 /■“ „(c) = — y exp{.4x'' - icx} dx 4- Rn{c), A = —/3 + ia^, where |/î„(c)| The proof of this Theorem follows from two estimations of different nature that we single out in two lemmas below. Let f = ip. Then I r°° I R n{c) = T - / i r i n - ^ x ) - e " ^ ) e - ^ dx 27T j Lemma 5.1.1 The following estimation holds true: |An(c)| < for some positive constant K Proof. Since G{t) = as É —0, we can choose Co > 0 so that |G(()| < for |(| < fo (5.3) 133 Taking ri > N and integrating by parts twice, we obtain: I A,(41 < |i J ’°ir(n-ix)-e‘^Ye-'=dx < 1 ^°° [(/"(n'il) - e'^)" dx Note that since by Lemma 4.1.1 limt_^±oo f{t) = 0 and since f is bounded, the non-integral term in both integrations disappears. Subdividing the integration range of the last integral at ±eon^, we write: i OC r rtoton*' n i (rCn-ïxl-e-^-")" dx= ^ {rin-x)-e-^r dx = I l -r I 2 / ' \x\> eon‘i ■OC J\x\>eoni J—font In 11 the integrand is estimated as: (r (n-ïx) - < 1(D"| 4 . |(e--^)"l (5.4) Clearly. n—2i W|2 I „ l ~ - | fin—11 rif\ Using Lemma 4.1.1 again, we find that each term is bounded by \n—iV—li r N \ I rff\ The second summand in (5.4) is bounded by an integrable function: |(e^^')"| < K x‘^-^e-^^ {{q -1)4- Jgx") It is easy to see that r o o / ^ (Cix’-2 4- dx < K n ^ e -^ ^ > J (on 9 134 Since / ‘^ is integrable and f , f" are bounded, we obtain: \h\ < Kn^~hl - 6Y < K (5.5) To estimate I 2 we first note that on [—eon?, /"(n“?x) = exp{.4x‘^+nG(n'?x)}. Thus, we consider I reo"’eon'< / i |/i"(a:)| dx ■font where hn(x) = ^exp{nG(n“?x)} — 1 j . In the following we will evaluate /i" and show that it is bounded by an integrable function. Denote nG(n”?x) by 0(x). Then /i^(x) = .4gx^~^e‘'^‘'(e® — 1) + and /i"(x) = Aqx‘>-'^e-^"(q - 1 + Agx*) (e® - l) + e^"e® [2Aqx‘>-^d' + &' + [ 9 ^ ] W e now estimate e®, 9' and 9" on the integration range. By the choice of eo in (5.3), if X 6 [-eon«, eon?], then we have |0 (x)| = |nG(n"?x)l < ^Ixl*» Similarly, using conditions on G' and G", we conclude that |0'(x)| = n^“?|G'(n“?x)| < ^\x \‘’~^ and Thus, |h%(x)| < P(x), where P(x) = K x‘^~^e~^'^{I + x ’) 135 - au integrable function on R. Thus, \h\ < K (5.6) for some positive constant K. It follows from (5.5) and (5.6) that IR.WI < g □ On the other hand, we can look at the remainder Rn{c) from a different point of view. Lemma 5.1.2 The following estimation holds true: ^ = 2ql2q + iy ^ ^ ^ ° Proof. Choose 6o as in (5.3). Let us subdivide the integration range at and ±.n'^, where 7 > 0 will be determined from the further arguments. Then 27TfL(c) = r (/"(n-iz) - e^^)e"‘“ dx = J —0 0 = [ ^ + ^ + r (r (n-ix) - e-^)e-^ dx = Ji+J2 + h J \x \> to n t ./n'><|xj Since / and are bounded by {1 —S) outside of a neighborhood of 0, we estimate the integrand of Ji as: Kr(n-iar) - e " ) e - ‘“ | < |/(n-Ji)|" + [e-^l < {1 - + le-^l 136 Since is integrable on R, by (3.12) \Ji\ < 2 f ^ dx < Kn-^e-^^> (5.7) J fonf To estimate the second integral, we note that on [—ean^\nG{n~^x)\ < ^\x\^. Hence. |/(n~?x)|" = and \e'’^''\ = e~'^^. Therefore, the integrand of J-i can be estimated as i(/"(n-^x) - e ^ )e --| < !/(»-#%)!" + |e^l <2 e 'T ^ Thus, we have, using (3.12) again \-h\ <2 J dx < exp | “ f ^ (5.8) To estimate J 3 integral, we consider the function nG{n~^x) on [—n'.n'^]. \riG{n~^x)\ < In order for the above bound to be o(l), we need the following condition on the parameter 7 : 0 < 7 < Thus. gnC7(n"«i) _ 2 and we obtain I./3 I < (5.9) which shows that our condition should be sharper: 0 < 7 < since we require J 3 to be 0 {n~^'-) for some h > 0 . 137 It follows from (5.7), (5.8) and (5.9) that \Rn{c)\ < K 4- j for some constant A' > 0 Choosing 7 so that 4- 7(g + 2) = - 7 ( 9 - 1 ) , we obtain 7 = ^(^V+T)' Hence \Rn{c)\ < where ^ ^ ^ n^^^' 2 q( 2 ç + 1 ) □ The above two lemmas lead us to the Proof (of Theorem 5.1.1). By definition c’„(c) = - ^ f /"(n"^x)e”‘“ dx J —OC .\dding and subtracting ^ exp{.4x^ — ica:} dx we can write 1 r°° ‘^nic) = — y exp{.4x’ - icx] dx + A„(c), where An(c) (r dx By Lemmas 5.1.1 and 5.1.2, (5.2) follows. □ Remark. Theorem 5.1.1 is an analogue of the Central Limit Theorem for the complex-valued probabilities in case p = 9 , and the inverse Fourier transform of exp{,4z'^} is an analogue of the normal distribution. 138 5.2 Asymptotics of the norm of n-fold convolution In this section we prove the following Theorem 5.2.1 [f the parameters p and q characterizing the behavior of ip at the origin are equal, then the L^-norm of n-fold convolution converge as n oo to the L^-norm of the inverse Fourier transform of the function g{x) = e '^ , A = —3 + ia^. Namely, = j du = 11^ ^g||n(R) + O , where ^ 2g(2g + 1) Proof. Since u = cn?, / \p'''^\u)\ du = n^ f dc f jw„(c)| dc J — oo J — oo J —oo Thus, we have: •OO 1 roc I r o o I / \ipn{c)\dc - — j / e^e"‘“ dx dc < -OO J —OO \ J —OO I r \ - e ^ )e * “ dx dc = r |&,(c)| dc —7T J —OC I •/ —OO J —oo The integral 1 rooroo I I roopoo .Ax^ %CT : r / / dx dc V —oOO o I •/•/ ——OCOC is I, ^-norm of the inverse Fourier transform of the function g{x) = e "^^. Estimate |i^(c)| dc. Splitting the integration range at ±n^, we write: f |i?„(c)|dc= f -h f |&,(c}| dc J —OO J lc|>n^ J — 139 Using (5.2) on |c( > n^. we get: r r°° 1 K / \R^{c)\dc t/*y(:/) G L^(R), A: = 0 , 1 , • • • , m, m > 3, (5.10) then the results on asymptotic behavior of i/^n(c) and will be improved (how ever at the expense of improving analytic properties of the function .^). Namely, we have the following two theorems. Theorem 5.2.2 If the function ç satisfies (4-1 ) j (5.10), (4-3) and (4-4)> is normalized so that a = 0. then the scaled n-fold convolutions iDn, defined by (5.1), will converge pointwise and in the L^-norm to the inverse Fourier transform of g{x) = exp{-4x^|. .4 = - J-I- ioq. Namely, 1 i/,i„(c) = — / exp(.4r‘* — icx) dx 4- Rn{c) 27T j where 140 Theorem 5.2.3 Under conditions in the above Theorem 5.2.2 , the -norms of the n-fold convolution converge as n oo to the U-norm of the inverse Fourier transform of the function g{x) — A = —0 + iUg. Namely, Ilf'" « t. = lf'"(wi = iij"-'î ||l.(»)+ 0 " h e r e ^ ■ 141 BIBLIOGRAPHY [1 ] Baishanski. B. M., Norms of Powers and a Central Limit Theorem for Complex- Valued Probabilities, Analysis of Divergence: Control and Management of Diver gent Processes, edited by W. Bray, Birkhauser Verlag, 1998. [2 ] Baishanski, B. M., Behavior of n-fold convolutions of complex-valued functions for large n, in preparation. [3] Baishanski, B. M. and Snell, M., Norms of powers of absolutely convergent Fourier series, Journal d'Analyse, 1998. [4] Gel’fand, B. V. and Shilov, G. E., Generalized functions, vol. 1 , Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1958. [5] Girard. D.. A General Asymptotic Formula For Analytic Functions, Dissertation, Ohio State University. 1968. [6 ] Girard, D.. The Asymptotic Behavior of Norms of Powers of Absolutely Conver gent Fourier Series, Pacific J. Math, 37 (1971), 357-387. [7] Gnedenko. B. V., The Theory of Probability, Chelsea publishing company. New York, N Y. , 1962. [8 ] Gnedenko. B. V. and Kolmogorov, A. N., Limit distributions for sums of inde pendent random variables, Addison-Wesley Publishing Company, 1968. [9] Hedstrom. G., Norms of Powers of Absolutely Convergent Fourier Series, Michi gan Math. J., 13 (1966), 393-416. [1 0 ] Hersh, R., A Class of “Central Limit Theorems” for Convolution Products of Generalized Functions, Trans. Amer. Math. Soc. 140 (1969), 71-85. [1 1 ] Hochberg, K. J., A Signed Measure on Path Space Related to Wiener Measure, Ann. Probab. 6 (1978), no. 3, 433-458. 142 [12] Hochberg, K. J., Central Limit Theorem for Signed Distributions, Proc. Amer. Math. Soc. 79 (1980), no. 2 , 298-302. [13] Krylov, V. .Ju.. A Limit Theorem. (Russian), Doki. Akad. Nauk SSSR 139 (1961). 20-23. [14] Kr\dov, V. Ju., Some Properties of the Distribution Corresponding to the Equa tion If = ( - l) " - ^ g # , Soviet Math. Doki. 1 (1960), 760-763. [15] Zhukov, A. I., Limit Theorem for Difference Operators. (Russian) Uspehi Mat. Nauk, 14 1959, no. 3 (87), 129-136. 143 IMAGE EVALUATION TEST TARGET (Q A -3 ) / f j /- 1.0 y â ■■ L à 2.2 c c in 2.0 i.i 1.25 1.4 1.6 150mm V 6>7 /1PPU ED ^ IM7IGE . Inc 1653 East Main Street ^ R octiester, NY 14609 USA Ptione: 716/482-0300 — — Fax: 716/288-5989 /; O '993. Applied Image. Inc.. All Rights Resetved '/