~, '" " f ' ~ (") ' ~" -~ ~ (, ~)

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1 N THE HARMNIC SUMMABILITY F FURIER SERIES BY J. A. SIDDII (Aahabad Universty, Atahabad) Received May ) 94 {Communieated by Prof. B. S. Madhava Rao). Let f(x) be a function integrabe in the sense of Lebesgue and periodic with period 2rr. Let the Fourier Series assoeiated with f(x) be 9 + Z (ah cos nx + b,, sin nx). A sequenee {&} is said to be summabe by harmonic means, " (.I),i Sn_k im,~ Z' (.2),.~ootog n,_-o k + exists. This method of summabiity which beongs to the r of generaized reguar N5rund methods*, was introduced for the first time by M. Rieszt who proved that every series summabe by harmonic means is summabe (C, ), > 0. The appication of harmonie summation to Fourier Series has been discussed by Juringius:~, Hie and Tamarkinw and Iyengar~. In this paper I prove the foowing theorem:-- THF.REM.mThe Fourier Series of the function f(x) is summabe by harmonic means at a point x at which # ~, '" " f ' ~ (") ' ~" -~ ~ (, ~) where g (t) = f(x + t) + f(x - 0-2f(x). I aro indebted to Dr. B. N. Prasad for his kind hep and advice in the preparation of this paper. * N6rund ().? R]esz (9). $ A proof of the theorem that th9 Fourier: Series is not summabe by harmon.:c rnr amost everywhem had been worked out by hito but remains unpubishr Sr Hi)r and Tamarkin (4), (5). w Hie and Tsmarkin (4), (5); f~ Iyengar (6), (7), 527

2 52 J.A. Siddiqi 2. In order to prove the theorem we sha require the foowing emmas: LEMMA I*. Ÿ < t < 7r, then! < A og =.---k qz i LEMMA II". For a vaues of n and x ] 2~ sin(k + ) t I t=,---/r ~9. 3. Proof of the Theorem. Le'. s,, (x) denote the n-th partia sum of the series (.). It is weu-known that i ( Sn(X ) --J'(x) ~- 2 dg(t) sin(n+~)t si 9 dt. t3") Making use of the formua (.2), we obtain 23 og n ~., k---+--i {sn'k -- f(x)} [' (n--k+9 -- o~ ~=0 j g (t).sin di -- ~ "2~* sin 9 : 9 sin (n -- k + 9 t = gtt)... ~, dt 2 ir og n t ~. o k +"~" sin 9 t T = fg(t)n,,(t)dt. 0 In order to prove the theorem we have to show that T f g (t) N,, (t) dt (I), (3 "2) as n ---~,o, when (.3) hods. We set Now 3 f $ r /g(,) Nn(t)dt = (/+ / +/) g(,) N,I (,) d, * For proof refer to Hardy and Rottosinski (3).? Gronwa (!)~ Titchmarsh (0), p. 440, = Li + L., + La. L, = f g (t) N. (t) at. m = (f Ig (t)!.ndt)

3 n tht Harmonª Summabiity of Fom'ª Seriss 529 -o o-t~ since N s (t) = 0 (n)~ for a t such that 0 ~< t ~~. Aso by virtue of Riemann-Lebesgue theorem and the reguarity "of the method of summatÿ we have f L, = f g (t) Nn Ÿ dt a = o (). (3.4.Lasty, in order to show that Lg = 0 () we require a suitabe estimate for the kerne N n (t)in the interva (-I n, ). Sinee N~ (0 -- x 2~r og i" sin 9 t k by making use of Lemmas I and II, we get ~,,~0~o[~ Hence a - k+--- cos n+ t k+-j' :., = f g (t) N. (0 at = 0 (f I g (0 I" I N,, (0 I a x x_ -(,o~f,,c,)i- n 6 a 3::~g + (o--o-ff I g 't) " } og ~ dt 0 dt II x. L { dt 9,og i-}/-~) + ~ f ~, (t) + This resut can bo estabished with the hep of the iaequaity [ sin nt [ ~ n I sin t [ and tho rdation 9 ~ ~ ogn. Se~ aso Hi9 and Tamarkia (5). n.

4 530 J.A. Siddiqi = (~~ D+ o (~,0~.,~>,) +o 0o~) 4-0 (ig-n; di... /) -4- r) ~,oi--n : :a:) t og 7- ~. :.o<,> + ot-7~-,,-j,)-io,,~~-,,,,,: Combining (3.3), (3.4) and (3-5) we see that (3-2) hods. theorem. 4. Iyengar* has proved the undernoted theorem:-- o o>. (3.5) This proves the If the series ~ ah is summabe by harmonic means and ah >--An-, 0< <, then the'series is convergent. Ir may be remarked that the theorem of this paper taken in eonjunction with the above mentioned theorem of yengar gives,',.nother proof of the foiowing we-known theorem of Hardy and Littewood " :~ i si (i) b,, Ah-' and () g(u) i du: (d---i)' then the FoU- rier Series converges to f(x). 5. [yengar~ has proved a theorem, viz., " tog ~ " Ir g (t) = 0 (-~gf ), then the Fourier Series of f(x) is summabe by -I harmonic means to f (x). This tÿ of Iyengar and the.theorem of this paper can both be proved indirecty by making use of a resut of Hie and Tamarkin. Hie and Tama'rkinw have estabished the foowing resuts:m Tn necessary and suffieient condition that the n-th harmonie mean of a Fo,irier Series sha converge to f(x) at a point x at whieh f I g (,),tu = o (t), u that im isg- sin,t. og i. g (t) (5.) ti --), oo * Iyengt.r (6), Theorem I. "" Hardy and Littewood (2). :~ Iyengar (6), Theorem II. w Hie and Tamark (5).

5 n the Harmonie Summabiity of Fourier Series If we assume Iyengar's condition, viz., ~">='~ ) t then the integra (5.), viz. I g(t)~ io--~- sin nt" og?" = og0~ $ 53 ~ 2. 3, ) 0. T. H. Gronwa.. G. H. Hardy and J. E Littewood..... and W. W. Rogosinski.. E. Hie and J. D. Tamarkin.. K. S.K. yengar N." E. N6rund M. Riesz E. C. Titchmarsh = 0 k,~ n d t/ =0(). n the other handif:we assumr the condition of the, theorem of this paper, viz.,! ~..>=;,~~,,>,.,,,,=o(~) then working on the ines of the ast paragraph of w 3, it can be shown that under this condition the integra (5.) is 0 (). In view of the great compication and diffir invoved in estabishing (5 -), the proof of the theorem as given here may be preferabe. REFERENCES n Math. Ann., 92, 72, Annai Scoua Norm. Super. Pisa., 932, 3, Proe. Camb. Phi. Soc., , Proc. Nat. Acad. Sci., (U.S.A.), 92, 4, Trans: Amer. Math. Soc., 932) 34) Proc. htd. Acad. Scg., (A), 943,, bid., 943,, Lunds Univer. Arsskrift, 99, N.F., ard. 2, bd. 6, no. 3, Proc. London Math. So. (2), 924, 22, Theory of Functions (xford), 939.