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(1.1) «-V (1.2) « 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—>+ <x>and m—►—<» the same is true of/(«). It is therefore suffi- cient to prove the theorem for «—>+ oo. |/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-> + <x>as <—>+ .' For any ¿>0 we have » 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 * ^ <rK £ | g(r) | £ ImIm-rK-^ - i-* £ /jr-«W £ g(r)/m_r r=l m—oo ffi=—oo r—1 to / 2 \—M » á e"x £ /«Ä--"«if8!2i-i f-1 ) = <r* £ /« | m | ! = *-*J8 m=-—oo \ X / —oo where 5=X)-»7m|w|!<9. Therefore Si^e-*B-»0 as /->0 + . ' 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.
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