A DISCRETE ANALOGUE OF THE WEIERSTRASS TRANSFORM1
EDWARD NORMAN 1. Introduction and definitions. In this note we consider a discrete analogue of the transform
,TÍ-g}*ttg(y)£yi /(*)=/_;
This transform may be considered as a convolution; accordingly our analogue will take the form
00 f(n) = £ k(n — m)g(m) —00
where the kernel sequence k(n) is to be appropriately chosen. The choice of k(n) is motivated as follows. One way in which the kernel e~y '* arises in the theory of the con- volution transform is by consideration of the class of kernels whose bilateral Laplace transform is of the form 6*ñ(i--)x- (1.1) «-V -tlak i \ aj When c = 0 we obtain the class of kernels whose analogue was con- sidered by Pollard and Standish [4] where the kernels formed a sub- class of the totally positive sequences. The totally positive sequences can be characterized as sequences having a generating function of the form
(1.2) «""■«"II 1 7:-7Z—r-77(1 - a,z)(i - bjZ-1)
(see [l]). The factor eaz+h'~ in (1.2) corresponds to the factor e" in (1.1). Letting a = b = v/2 we have the familiar expansion
eC/2) (H-*"1) = Y,Im(v)zm
Received by the editors February 10, 1959 and, in revised form, October 22, 1959. 1 This work is part of a doctoral thesis done under the direction of Professor Harry Pollard at Cornell University. * See Hirschman and Widder [3] for the theory of the Weierstrass transform.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use596 DISCRETE ANALOGUEOF WEIERSTRASS TRANSFORM 597
where Im(v) is the modified Bessel coefficient of the first kind. We take the sequence {J„(l )}"__„= {j„} "«, as the kernel of our transform
(1.3) /(«) = 23 In-mg(m).
Before proceeding further we list some elementary properties of the functions In(v). (v/2Y (i) J»(») = £ w = 0, ±1, ± o s!(s + w)! (Ü) Ir.(v) = I-n(v) ; In(-V) = (- l)»In(v), 2(l/2)'»i ^ , „. ^ 3(1/2)1-1 (iii) —j—¡—- S InW S —¡—r—t
(iv) In(v + V') = ¿ In-m(v)Im(v').
If we take g(m) = 2|m||»z| !/(1-)-»í2) it is clear, by (iii), that (2.13) converges only for the value w = 0. We therefore define the transform (1.3) only when it converges for all w. Define the operators ô and 5_1 by the equations 5/(w) =/(w-l), 6-1/(w)=/(w+l). Then by analogy with the theory of the Weierstrass transform we may expect (1.3) to be inverted by the operator e~ai2ns+s~ ' inter- preted as (1.4) [«-»'»<«+|-1>]/(n) = lim £ (-l)mImf(n-m)/T(l + im). i-»o+ _„ The method of summation adopted in (1.4) is a regular method introduced by Mittag-Leffier (see [2; p. 72]). The motivation for choosing Mittag-Leffier summation as the "natural" method for the inversion operator is indicated in the table below.
Discrete
Kernel Im 1/m!
Convergence factor l/T(l+f|m|) i->0 +
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 598 EDWARD NORMAN [August
Unfortunately the operator (1.4) will not serve to invert all con- vergent transforms (1.3). For example let ' 2mm\ m > 1, g(m) = mlogm 10 mal. Then, if «>0 say, we have " 2mm\ " 2mm\
/(») = Ei I«-»—-'mloe m > E i 2|n-™l | « - w | !wlog ' " 2"w! " mn > Et-TT——„ (m — n) \mlos m > E „ wlog m
If in the last sum above we consider only the term with m= [enl2] we obtain
/(») > 0(e"2/4) » -» + oo ;
and (1.4) does not converge for any value of t. In view of the above we restrict attention to the class of determin- ing sequences g(m) which satisfy the condition g(m) = 0(\m\ !xl"*l) as |m|—>oo, where x is strictly less than 2. We abbreviate this re- striction by writing g(m) EX. 2. The inversion theorem. We first find the order of the transform.
Theorem 1. // g(m)EX then /(«) = 22"„ In-mg(m) converges ab- solutely for all «, and (2.1) /(«) = 0(2l"'| «|!(2/x- l)-l"l) | «| ->°o. Proof. The first statement is obvious. To prove the second, note that since the order conditions on g(m) and Im are the same for «z—>+
|/M| £¿7»—U(»»)| +iln-m\g(m)\ =S' + S. —oo 0
Since «>0 it is clear that 5'^const. Y1ô(x/2)m/mn—»0 as «—»<».
» (x/2)mm\ °° 5 g const. 2" Yj -■ = const. xn Y (m + 1) - ' ' (m + n)(x/2)m. n (m — «) ! o
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use i960] DISCRETE ANALOGUEOF WEIERSTRASS TRANSFORM 599
The formula
CO E (m + 1) ■ • • (m + n)ym = w!/(l - y)n+1, y < 1 0 may be proved by induction. Hence we have
xnn\ S < const.-= const. 2"w!(2/x — 1)_". (1 - x/2)« Theorem 2. If g(n)^X and f(n) = E-» In-mg(m), then
00 (2.2) lim 2~2(-l)mImf(n-m)/Y(\ + t\m\) = g(n). t—0+ -«,
Proof. There is no loss of generality in taking w = 0. For any i>0
CO £(-i)»jm/(-w)/r(i + i|w|) —00 00 00 = E (-l)mIm/T(l + t\m\) E I-m-rg(r) m =—oo r=—oo
00 00 00 = Z g(r) E (-l)"'J-m_rJm/r(l + i|W|)= E g(r)p(r,l). r=—co ffi=—co r=—oo The interchange of summations is justified by the estimate of Theo- rem 1. Since CO CO m(o, o) = E (-i)raJ-m(i)Jm(i) = £/-«(l)/m(-l) = Jo(0) = 1 —00 —co (see iv) the theorem will be established if we show
CO (2.3) lim 2Zg(r)p(r,t) = 0, <-o+ i
(2.4) lim 2Zg(r)p(r,t) =0. <-*>+ -co We will prove (2.3) ; the proof of (2.4) is entirely similar. As a first step let us replace l/r(l-f-i| m\ ) by the contour integral 1 1 c(0+) e" —j-¡— =- I — u-'imldu T(l + t\m\) 2-kíJ _„ u where the path of integration is as shown, the circle being of radius greater than one.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 600 EDWARD NORMAN [August
arg
arg «=— ir
Define the function K = K(t) =21"(2/x-1)"1", so that K-> +
» 00 00 ~ (0+) gu 2« E «W/*(r,0 = E «to E (- Dm/m/n,-r 1 —8-«l"ld8 1 r«*»l m——oo ** —oo W = EgtoE(-Dm(---) f (•••)*» + E«WE(-!)"•(•••)f (•••■)* + EiWE(-!)"(••■)f (•••)* = 5 + 5! + Si. The path C runs from u=—K(argu=—ir) round the circle and back to u— —K(ar% u = ir), G runs from -»to —if (arg u= —«•), C2 runs from —K to — « (arg « = 7r). We show now that lim,_0+ Si = 0; the argument showing Iim(<0+ S2 = 0 is the same.
\$l\ S £ I «to I £ W»-» f — 8-«l»l¿8 r-1 m—oo »Ci *
^
' It is essential to the proof that K—>°o. This is not the case when *S2/3. But in that event the sum (1.4), with t = 0, converges, and the proof that (1.4) inverts (1.3) is trivial. We therefore assume from now on that *>2/3.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use i960] DISCRETE ANALOGUE OF WEIERSTRASS TRANSFORM 601
The proof has now been reduced to showing that S—>0 as t—>0+ . To this end we make use of the formula [5, p. 441] 2 ("I*f'l* Jm(l)Jr_m(l) = — I Jr(2 cos 8) cos (2m - r)6dd. IT J on We obtain, setting u~' = w,
IT oo oo *%x/2 i* eu S=2~2g(r) E (-l)m I Jr(2 cos 8) cos (2m - r)8d8 j —w^du 2 r=l m—oo / 0 » C U
CO - E g(r)v(r, t). r-l Estimate of v(r, t). Since | w\ <1 along the entire path C, we may write
/' e" crl2 ! °° ) — | Jr(2 cos 8) < E (-1)1"«"» cos (2w - r)ô> ¿0¿m. C U J o \ m--co J " 1 - W2 2_! (— í)mw^ cos(2w - r)8 = cos(rô) ■ 1 + 2w cos 20 + w2 We now have "l2 IT(2 cos 8) cos rö /' -e" (1 - «,*)I rT/2 - dddu. 0 « J o 1 + 2w cos 20 + w2 The integral 1*l* IT(2 cos 8) cos r0 /. r/2 de 0 1 + 2w cos 20 + w2
may be estimated as follows. Replace Jr(2 cos 8) by the sum (i) and interchange the order of summation and integration to obtain
co { p ' T/2i/2 (cos)| 8r+2s cos r0 (2.5)•5) Qr(w) = E}_, -77--77—T- I 7-;d8. ,=o i!(f + s)[J o 1 + 2w cos 28 + w2
The integral appearing in (2.5) is now estimated by contour integra- tion. It is easily verified that
/.x/2 1 /* 2t (■■■)dd = - ( -. - )dêt o 4 / o and so the standard substitution cos 0 = 1/2(2 + 1/3), z = ei6 yields
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 602 EDWARD NORMAN [August
cTlî r (z2 + i)r+2s(32r +1) i2*+2>+3I ( ■• • )dd = I-dz J o J |,|=i z2^2'-1^4 + (1 + w2)z2 + w)
= I pi(z)dz + I p2(z)dz J M=l J \z\-\ where
(Z2 + 1)H-* Pi(z) = Z2,-1(WZ4 + (1 + W2)32 + W) (z2 + 1)--+2' P2(z) = 22r+2S-l(w24 _|_ (1 _|_ w2)z2 _|_ w)
Both pi(z) and p2(z) have poles at 2 = 0 and at the zeros of wzi + (l+w2)z2+w, that is, at 2= +iw112, ±i/w112. The points ±iw112 lie inside the unit circle, while +i/w112 lie outside. Estimate of J\z\=\pi(z)dz. Let a be a real positive number such that
1 | , 1/1 IV'2 a < inf — | w1/2 -1(1-1)' uec 2 2 \x 2/
/I Pi(z)dz\ = If I Pl(z)dz\+I .I Respi(z) .|2_±1v/2 |zl=l I J |i|=a
HI r+2s Res pi(z) !_+<„»» = 2 | w' | | 1 — w21
(a2+ 1)--+2« /pi(z)dz Pi(z)dz = 2ira2-:—:-Mi(w) 1*1-«i,
where Mi(w)=sup\z\=a | wz+(l+w2)z2+w2| _1. Afi(w) certainly ex- ists since the choice of a assures that wz4 + (l +w2)z2+w2 is bounded away from zero for |z| =a. Now, fix a so small that x(a2 + l) <2. Estimate of f\z\=ipz(z)dz. Let ß be a real positive number such that 2 ß > sup 2/ w1'2 = -, »eC ' ' /l IX1'2 VT~T/
/I p2(z)dz = I If p2(«)az li+ | Res p2(z) i|z„±nww,
w r+2» Res p2(z) |z_±i/u,i'2 = 2 I W « 1 - W2
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use i960] DISCRETE ANALOGUE OF WEIERSTRASS TRANSFORM 603
(ß2 + iy+2' p2(z)dz g 2irß2-M2(w) 1/ «2r+2« where M¡(w)=sup¡z¡^\wzi + (í+w2)z2+w\~l. The choice of ß as- sures that M2(w) exists. Now fix ß so large that x(ß2 + l)/ß2<2. We are now ready to give the estimate of v(r, t).
/' I —eu I | 1 - w21 | Qr(w)| | du I c\ u \ where « 1 ill- w\r+23 (a2 + ly+^Mt Q(w)s E-\t-t-,—!-r + x-- Zl sl(r + s)l2r+2> 11 w\°\ 1 - w2\ a2'-2 (2.6) (02 + 1)^+2^^ _|_ 2v-} . 02r+2a—2 I
The estimate (2.6) enables us to write
CO E I g(r) | | v(r, t) | (2.7) * f \ — \\ l-w2\ j¿ \g(r)\ \Qr(w)\\du. J c\ u \ \ i ; The proof is now completed by breaking (2.7) into three parts cor- responding to the three sums in (2.6). Call these parts J\, T2, and T3. We give the details showing J\—>0 as i—>0+ . The details for T2 and Tí are similar, and somewhat simpler after observing that Mi(w) and M2(w) are bounded for w£C (again by the choice of a and ß).
— | 1 - w21 /'«Hic\ u I i í A i i A 1 I l - w\r+2" ) • < E g(r) E - T^-r-,-!-rt fa- il Í-Í sl(r + s)l2*+2° \w\°\ 1 - w2\ ) Since (r+s)\^r\s\ and since 22s(s!)2^(2s)! we have • 1 | 1 - w Ir+23
tZ s\(r + s)I2r+2' \w\'\l-w2\ I 1 - wV A I 1 - w\2s 1 . e\l-w\l\w\l | 1 - w21 rl2* tZ (\w\ i'2)28 (25) ! | 1 - w2 \r !2r so that
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 604 EDWARD NORMAN
/' \ -Lu-.i/h'^^J-Le" \ °° 1 — wW\ \r \d, cl « I i 2rr!
/' I eu I „„ | 1 — w-' I du] cl M I 1 — x/2 1 — M' where Ai is a constant. Break the path C into two parts, C* and C* as shown: C* i i _l__
v -K -AT
Given €>0 choose / so small that there is a fixed positive number N /. elu'-l||u|« -!-1- I 1 —u~' I ¿M < - c. 1 - x/2 | 1 - «"' | 2 and also sud | 1 1 - N-* PC (2.9) <-(2 VJc'l f l-U'-1'"""2-Î-rl*»lÜ « I 1 - x/2 | 1 - «-'I '/ This can be done by virtue of the fact that K—->+ » as /—»0+ and the fact that the integrals in (2.8) and (2.9) are seen, by elementary estimates, to be bounded independently of t. Thus Ti License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use