ON PRODUCTS OF SUMMABILITY METHODS OTTO SZÁSZ

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1 ON PRODUCTS OF SUMMABILITY METHODS OTTO SZÁSZ 1. At the recent Internatinal Cngress I. M. Sheffer asked me the fllwing questin : Given a sequence {sn} ; frm the sequence {a^} í the Cesàr means f rder a (a>0) crrespnding t {sn}. If {sn} is summable Abel, is it true that (^J is als summable Abel? A mre general questin is: Suppse that A and B are tw regular summability methds fr sequences {sn}. Dente by AB the iteratin prduct which assciates with a given sequence the A transfrm f its B transfrm ; when des A summability imply AB summability? We shall shw in 2 that the answer is affirmative when B is(c, a) and A is Abel summability.1 In 3 we generalize this result t Laplace transfrms and Riesz summability. In 4 we discuss the iteratin prduct f Cesàr and Brel summability, and als Euler and Brel summability. 2. Let/(x)= X anxn = (l x) ^snxn. Abel summability f ^an t s is We define s" by the identity A lim sn = lim (1 x) Z snxn = i. Z->1 and -ft by f(x) = (1 - x)" X)s«*n> 0 (1 - X) = X)7»Xn, s that 0 OK «r V(a + n+l) (2.1) 7n = Cn+a.n = - - ' «ir(«+ 1) We nw have (2.2) <r = > sn = Z svyn-v yl v= a Presented t the Sciety, April 27, 1951 ; received by the editrs February 7, Fr the case a = l see [4, p. 189]; [3, p. 258]. Numbers in brackets refer t the literature at the end f this paper. 257

2 258 OTTO SZÁSZ [April Furthermre (2.3) = af p»(l- p)^dp, The transfrmatin 00 Cl i 2~2 Cx" = a I (1 - p) 2 Sn(px)"dp J 0 = a f (1 - p)b~(l - pxf ""'/(P*)^-2 ' yields px = 1 t~l, xdp = t~2dt» fy / i/-«) / / l\«-i 0 X ^i \ X / Putting x = 1 (y +1)_1 (y >» ) yields Nw set /=«+!; then / 1 \ «f y+i I t - n «-1 = H1+7)/. y." v r) /(1"r '-!) í. (2.4) y+lv V y+l/ where <p(u) =/(l (m + 1)_1). Nw / IX"-1 a C / «Xa-1 = (l + ) (l--) 4>(u)du, \ y/ y \ y/ y/ y a CyI u\a-1 I 11-) 0(m) * W \ y/ is the integral (C, a) transfrm f the functin <b(u) ; denting it by (C, a) we have frm (2.4) * The same frmula appears in [2, p. 200]. See als E. C. Titchmarsh, The thery f functins, Oxfrd, 1932, p. 242, example 8.

3 1952] ON PRODUCTS OF SUMMABILITY METHODS 259 A(C, a){sn} «(l + j (C, «){*(«)}, frm which ur assertin fllws, as the right side tends t s fr y > c. 3. Abel summability has been generalized t Dirichlet series 00 D(t) = XI ane-y"', where 0 g X0 < Xi <, X -». If the Dirichlet series cnverges fr t>0, and if D(t) >s as /»0, then we write D\ ^Tlan = s. The methd is regular. T Abel summability crrespnds the case X = «, e~' = x. Furthermre Intrducing the stepfunctin /» Xn+1 D(t) = tyssn I J\ e-"'du. is, fr \»Sa< X +1, «= 0, 1, 2,, (3.1) s(u) = < lo, fr 0 < î< < X0, if X > 0, we get (3.2) D(t) = t f s(u)e-utdu. J 0 If sn»5, then s(u) *s, u», and limí<0 -D(/) = * defines the generalized limit f s(u). In general (3.2) is a regular transfrm f the functin s(u), the Laplace transfrm fr which we write L{s(u), t}. The (C, a) means f s(u) are Ca(x) =" ax-" I (x - u) -ls(u)du, a > 0. w In the case f (3.1) if X gx<x +i, then C0(x) reduces t / K 1 /» X»+l / I \ i?a(x) = aara < X] if I (* u)a~1du + sn I (x u)a-ldu> \ J\, J\n * n = x~a X] (x - X )a. These are Riesz's typical means f rder a (see [l, chap. 4]). If Ra(x) *s, then the series Xa«s called summable t s(k, a). It is knwn that [l, p. 39] if ^a is summable (X, a), then it is summable

4 260 OTTO SZÁSZ [April D\ t the same value. Similarly if Ca(x) *s (x»»), then L{s(u), t} -*s (t-*0). We have [l, p. 39] D =,, ^ I ^(^e-^du, T(a+ I) J where sa(x) =afx(x u)a~1s(u)du. Nw Ca(x) = x~"sa(x), We have s that fcc+l f* 00 D(t) = - T(a+l)J I uacju)e-utdu. /I p"-le-u"dp, a > 0, 0 /. 00 Ca(u)e-Utdu = - I uaca(u)e-ui I pa-xe-u"dpdu T(a) J0 J t r f =- I p"-1 I sa(u)e-u<-t+>l)dudp T(a) J «^ = a«i p""1^ + p)-«-1z)(/ + p)dp. ^ Suppse that L{s(u), t}-*s as i-»0. Nw aí/ V_1(í + p)~a~ldp = L /> 00 p«-i( - _ p)-«-i{d(t + p) - s}dp. But it is easily shwn that this expressin tends t 0. Thus the therem : If then L{s(u), t} >s as t * 0, L{Ca(x), t} ->j as i-»0.

5 i9c2] ON PRODUCTS OF SUMMABILITY METHODS We nw cnsider the prduct f Brel and (C, a) summability. The Brel transfrm f a sequence {s } is 00 s B{sn, x} = erxj^ x\»! We say that (B) lim sn = s, iilimx^x B{sn, x} =s. The prduct B(C, a) is the transfrm Emplying (2.1) and (2.2) we have _ x" B{ffn, X} = e "E,ffn nl *.n /*n n Z «; = r(«+ l) X) r:,,. Z) *.7 -. = w' n= F(a +» + 1) «- r(a + 1) " x" " T(a + n - v) r(a) _ Y(a + n+ 1) _ ^ (n - v)\ = a X] sv X *" T(a + n v) (n - î>)!r(a- - n + 1) The interchange f summatin is legitimate if the duble sum is abslutely cnvergent. Frm (2.3) v\t(a + n v) Çx,--7T = P^"-" P)vdp, T(a+n+l) J " T(a + n - v) If1 " xn V xn - -= I pa_1(l - P)v jl, -pn~vdp ZI (n - v)lt(a + n+l) vl J 0 = (n - v)! It fllws that ±\s\±*- Tia+n-v) Zu I "v I /_, x,= n=» («-») r(a + n+ I) x" r1 = I p ->(l - p)ve"xdp. vlj 00 ~.t) /» 1 «.»' ' < I p^-v1! Z *. )dp- J \»= s!/

6 262 OTTO SZÁSZ [April By assumptin the sum ^ sn\ xn/n\ is an entire functin, the duble sum n the left is cnvergent fr all x>0; this prves that 22, n x"/w! is an entire functin, and ~n ~.v /» 1 Z "n = <* Z Sv n=»! «~» ' p"-l(l - p)"e"xdp r1 " x"(l -p)» = a I pa~xe"x 22 s -dp J v\ Finally r = a pa-1e"xe'-1->''>xb{sn,(l - p)x}dp. J 0 B{a", x} = a I p^ßjin, (1 - p)x} p ^ = xj f (i- ) X x/ S {*»,*}*. J3(C, «){* } = (C, a)5{i /}. It fllws that (B) lim s = s implies (B) lim <r =.ç. Fr a = l this result (with a simple prf) was cmmunicated t me by J. Barlaz at the summer meeting, 1949, in Bulder, Clrad. We finally cnsider the prduct f Brel and the generalized Euler transfrm. The Euler transfrm Er is defined by n <t>n(r) = EC,,/(1 - r)»-»,. r=0 It was shwn by K. Knpp that this transfrm is regular if and nly if r is real and 0 < r ^ 1. Nw ~n x ~n n B{<l>n(r), x} = e~x2z <t>n(r) - e ~i]c.,,f(l - r) s, n\ =0 ft I v-0 JL rv " xn(l - r)n = e-x2z Sv (1 - r)- Z -;-7 *_»! n-» («-»)! A (rx)v.. = e-z2^ Sv-eO-D«= b{s, rx\.»! The interchange f summatin is legitimate, the duble series being

7 195*1 A NOTE ON INDEFINITE INTEGRALS 263 abslutely cnvergent. Hence B{tpn(r), x} = B{sn, rx}. It fllws that (B) lim sn = s implies (B) lim <bn(r) =s. References 1. G. H. Hardy and M. Riesz, The general thery f Dirichlet's series, Lndn, E. Kgbetliantz, Sur la smmatin des séries divergentes par les myennes simples et dubles, Ann. Écle Nrm. (3) vl. 42 (1925) pp Ott Szász, Verallgemeinerung eines Littlewd'sehen Salzes über Ptenzreihen, J. Lndn Math. Sc. vl. 3 (1928) pp A. Zygmund, Remarque sur la smmabilité des séries de fnctins rthgnales, Bulletin internatinal de l'académie Plnaise des sciences et des lettres, Classe des Sciences Math, et Nat. Série A (1926) pp., University f Cincinnati A NOTE ON INDEFINITE INTEGRALS J. D. HILL I. Thrughut the paper /(x) will dente a given functin, realvalued and Lebesgue integrable n the interval X = (0^x^l). We dente by E the generic measurable subset f X and intrduce the fllwing definitins. (1.1) F(E)=F(E;f)=fEf(x)dx. (1.2) B*(a)=B*(a; f) and B*(a)=B*(a; f) are, respectively, the greatest lwer and least upper bunds f F(E) taken ver all sets E f measure \E\ =a (O^agl). Regarded as a functin f E, F(E) is called a generalized indefinite integral f f(x) [4] r simply an indefinite integral f f(x) [lj. In this sectin we develp the principal prperties f the functins B*(a) and B*(a), and then btain as a main result (see (1.9)) the fact that the values f F(E) fr =a cmprise the clsed interval frm B*(a) t B*(a). This is an extensin f the knwn fact that F(E) assumes all values between its ptimum bunds, where n restrictins are placed n the measures f the sets invlved [4]. In the secnd sectin the results f the first are applied t the prblem f defining a mean value fr F(E) as E ranges ver the measurable subsets f X. Received by the editrs April 26, 1951.