ON THE ABSOLUTE CONVERGENCE OF A SERIES ASSOCIATED WITH A FOURIER SERIES. R. MOHANTY and s. mohapatra

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1 ON THE ABSOLUTE CONVERGENCE OF A SERIES ASSOCIATED WITH A FOURIER SERIES R. MOHANTY and s. mhapatra 1. Suppse/(i) is integrable L in ( ir, it) peridic with perid 2ir, and that its Furier series at / = * is 1 " (1.1) a0 + 2~1 (a» cs nx + n sin nx) = 2~1 A 2 i and its cnjugate series at t = x is (1.2) 2~1 (n cs nx a sin nx) = 2J Bn. i We shall be cncerned in this nte with the series (1.3) i^i» where (1.4) 5» = 4* i and 5 is an apprpriate number independent f». We write (1.5) <p(t) = {f(x + i) + f(x - t) - 2s}/2, (1.6) <j>a(t) = T" I it - u)*-l<t>iu)du, a > 0, and fw = *(/)» (1.7) ^(i)= {/(* + 0-/(*-0}/2. The rdinary Cesàr summability f the series (1.3) was first studied by Hardy and Littlewd [S] wh bserved that the relatin f (1.3) t the integral J(<P(t)/t)dt is very similar t that between the allied series and the integral f0(y /(t)/t)dt. Zygmund [lo] gave a necessary and sufficient cnditin fr the cnvergence f the same series (1.3). The bject f the present nte is t study the abslute cnvergence and abslute Riesz summability f (1.3). We prve the fllwing Therem 1. If (1) <bx(t) lg k/t is f bunded variatin in (0, it), Received by the editrs Nvember 28,

2 1050 R. MOHANTY AND S. MOHAPATRA [December (2) \(piit)\/t is integrable in (0, tt), (3) {«M } is f bunded variatin, 5>0, then (1.3) is abslutely cnvergent. Therem 2. If the series (1.3) is abslutely cnvergent then fô(\^t+»it)\/t)dt<», 5>0. Therem 3. If \(p(t) \ t is integrable in (0, it) then the series (1.3) is summable \C, h\, ô>0. In Therems 1 and 1(a), and the lemmas which fllw, # is any number greater than ir. 2. Fr prving Therem 1 we first prve Therem 1(a). If (1) (pi(t) lg (k/t) is f bunded variatin in (0, it), /t is integrable in (0, it), then the series (1.3) is summable, 0<a<l. (2) IM) 2?,e«M We require the fllwing lemmas fr the prf f Therem 1(a). Lemma 1. If q>(t) lg (k/t) is f bunded variatin in (0, ir), then the series (1.1) is summable \R, e"a, l, 0<a<l. Lemma 2. If \ /(t) lg (k/t) is f bunded variatin in (0, ir) and ip(t)\/t is integrable in (0, 7r), then the series (1.2) is summable R,en", l,0<a<l. Lemma 3. If <b(t) lg (k/t) is f bunded variatin in (0, tt) and \<b(t)\/t is integrable in (0, it), then the series (1.3) is abslutely cnvergent. Lemmas 1 and 2 are knwn [ó], whereas Lemma 3 with relaxed hypthesis can be prved by Tauberian argument, but we give the fllwing direct prf which will suffice fr ur purpse. Prf f Lemma 3. Since <f>(t) is f bunded variatin in (0, ir), we have s ~~ s 1 (*T sin fit --, *w dt + (w~2) =l» + (w~2) sayn 2t n J I Nw -1 -i 1i rt L = --I 2t n J d^unit); n integratin by parts, where Unit) =Jj sin (mm/m) du. It will be enugh t prve that HI '» I < Nw,

3 I9J6J A SERIES ASSOCIATED WITH A FOURIER SERIES 1051 EKi = ^f*l *! E- IT J» \Unit)\ J 0 V náfcí-1» * n>t<-l»v 0 lg- < 00, t /' T & using fairly bvius inequalities fr Unit), viz. We nw begin t prve Therem 1 (a). Sn - s If sin (n + l/2)t -= - 0(0 «WIT*' 2 sin (i/2) 1 / T sin»/ 1 rt cs nt I 0(0 ct it/2)-dt -\-I 0(0-dl 2irJ n 2irJ» 1 ( + ft,), say. IT it On integratin by parts where /' r 0i(O / </2 y sin»< dt / \sin(í/2)/ n H--I 2 J <bi(t)t ct (í/2) cs»/ i /' * 7(0 -Ä+ / «2 J sin «/ dt-\-i If G(t) cs nt dt = Pn + Qn, say, / i/2 V F(0 = 0i(O (..,. ) and G(0 = H>x(t) ct (i/2). \sin (</2)/ 23! ^n I is cnvergent by Lemma 3 with F(t) in place f 0(0 and 2~liQn is summable \R, en", l, 0<a<l, by Lemma 1 with G(0 in place f 0(0 and hence 2~lan is summable R, en, 11. On integratin by parts, cs»x 1 r * ßn = Hit)-I! $(0} sin nidi, = yn + Sn, say. n 2irJ

4 1052 R. MOHANTY AND S. MOHAPATRA [December Hi* is summable R, e"a, 11 by knwn results [7] and the discussin f R, en", l summability f XX is similar t that f the cnjugate series H^n with td>i(t) taking place f \ /(t) and therefre by Lemma 2 it fllws that 2X is summable \R, en", l if J\d{t( >i(t)}\ lgk/t <», which is satisfied under the hypthesis f Therem 1(a), since /' * k Cr k CT k \d{td>i(t)} lg â I ^1.(0 lg dt+\ t\d<pi\\g 0 t J t J I and bth the integrals n the right are cnvergent by the hypthesis f Therem 1(a) and thus the prf f Therem 1(a) is cmplete. Cmbining Therem 1(a) with the fllwing knwn therem [9], prf f Therem 1 is cmpleted with (sn s)/n in place f a. Therem A. If (1) H0-* «summable \R, en", l, (2) {m1-"«^} is f bunded variatin, then H\an\ is cnvergent. Under hyptheses (1) and (2) f Therem 1 the series (1.3) is summable R, e"a, 11 and it wuld be abslutely cnvergent, if {«1-a(sn s)/n} is f bunded variatin, i.e. if {(sn s)/n"} is f bunded variatin, i.e. if {s /wa} is f bunded variatin. Nw (Sn) S Sn-l Sn ~ Sn-1 ( 1 1 ) A < > =-=-\- S _l <-} in") na (n \)a na {na (n l)a) Therefre él t. f i n n" e a(-) = H ^+H ia ((«l)a na) ' «-i {t-v-1} I \«vl na \(n - l)a n") = E + (i) na by hypthesis 1 f Therem 1 which als ensures C\ summability f H^n and a frtiri that f H^-n/n". Hence y,.4 In" will be cnvergent by the apprpriate tauberian therem abve if the sequence {n(an/na)} is f bunded variatin, i.e. if {»1_M } is f bunded variatin, i.e. if {«M } is f bunded variatin, writing a = \ S, which is the cnditin (3) f Therem Prf f Therem 2. We have [l ] 0i+s(O = (l + ô) E^Yi-h^O where y,(x) =/ (l u)'~l cs uxdu. Nw

5 1956] A SERIES ASSOCIATED WITH A FOURIER SERIES 1053 /' ' I 0i+s(O I ^,, CT i i dt I J t áaz <^ n by using apprpriate inequalities fr Ayi+a(»0 [2]. Remarks. It is wrth remarking here that if fix) belngs t Lip a, 0<a<l, i.e. f(x+h) f(x) =0{ \h\ "} unifrmly as h *0, then 2~2\sn f(x)\/n cnverges unifrmly in x. Fr it is well knwn [lo] that when f(x) belngs t Lip a, then sn f(x) = O {lg»/»a} unifrmly in x. But the Furier series f a functin belnging t Lip a is abslutely cnvergent when a>l/2. Again the Furier series f a functin can be abslutely cnvergent at a pint withut the series (1.3) being abslutely cnvergent at that pint. Fr instance, if 0(0 = (lg (&/M))-1 m (0> tt) and defined by peridicity elsewhere, it is knwn that [8] An = 0\ (l/«(lg w)2)} and hence.<4b <, i.e. the Furier series f 0(0 at t = 0 is abslutely cnvergent. But J \<fa(t) /t is nt cnvergent fr any X and hence the series can nt be abslutely cnvergent by Therem 2 f the present nte. 4. The prf f Therem 3 can be effected in the same manner as that f a knwn result [3]. Frm this therem it is clear that ] C, S summability f (1.3) is a lcal prperty f its generating functin whereas its abslute cnvergence is nt s, [4]. References 1. L. S. Bsanquet, Prc. Lndn Math. Sc. (2) vl. 31 (1936) p , Le. cit. p , J. Lndn Math. Sc. vl. 11 (1936) p Bsanquet and Kestelman, Prc. Lndn Math. Sc. (2) vl. 45 (1938) pp Hardy and Littlewd, Prc. Lndn Math. Sc. (2) vl. 24 (1926) p. 238, Therem R. Mhanty, Prc. Lndn Math. Sc. (2) vl. 52 (1951) p. 229, Therem B. 7. -, Le. cit. p. 315, Therem , Le. cit. (2) vl. 51 (1949) p , Prc. Lndn Math. Sc. (2) vl. 51 (1949) p. 188, Therem C 10. A. Zygmund, Trignmetric series, 1935, p. 61. Ravenshaw Cllege, Cuttack, India