ON THE HARMONIC SUMMABILITY OF DOUBLE FOURIER SERIES P. L. SHARMA

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1 ON THE HARMONIC SUMMABILITY OF DOUBLE FOURIER SERIES P. L. SHARMA 1. Suppose that the function f(u, v) is integrable in the sense of Lebesgue, over the square ( ir, ir; it, it) and is periodic with period 27T in each variable. Let 1 <p(u, v) = <px(u, v) = [f(x + u, y + v) + f(x + u, y - v) 4 + f(x u, y + v) + f(x u, y v) A-s]. Definition. The double sequence {sm,»} is said to be summable by Harmonic means, or summable (H, 1, 1) if 1 m n c,, 1 ^ * t~-\ m l,n k lim -2-i 2-i -»,-.«,»-.«. logwlogw i=o k~0 (I + 1)(* + 1) exists. This is a particular case pn = l/(n + I) of Norlund summability of a double sequence as defined by Herriot [4], Hille and Tamarkin [5] have proved the following theorem on the Harmonic Summability of Fourier Series: Theorem A. The Fourier Series of the function f(x) is summable (H, 1) at the point x at which f' [ t 0i(O = I I 4>(u) \ du = o - where <p(t) =f(x+t)+f(x t) 2f(x). Jo 1 log An easy proof of this theorem is given by Prasad and Siddiqui [6], We shall prove the theorem: Theorem B. If furv\ i [ uv $(m, v) = I ds I cb(s, t)\ dt = o -, J o J o 11 log log u v /" I r [ u ' dl\ I <p(s, t)ds = 0 -, o IJ o 1 log u Received by the Editors October 3,

2 980 P. L. SHARMA [December /" ds\ I I /" <b(s, t)dt = 0 f - V 0 I J 0 1 log V then the double Fourier Series of function f(u, v) is summable (H, 1, 1) to the sum s at u=x and v=y. This theorem is a generalization of Theorem A for double Fourier series and also is analogous to the theorem of Chow [l] for summability (C, 1, 1) of the double Fourier series. 2. We require the following lemmas: Lemma 1. If0<t<ir, This is known [3]. then» cos (k+l)t ( 1 \ Z-kTi-< AV+*7)' Lemma 2. For all values of ra and x This is known [7]. Lemma 3. For t such that O^t^ " sin (k + l)t 1 E- =» + 1 k-o k+l 2 l/n 1» 1 sin (n - y + l/2)t I kn(t) = -E 7 - ~- = O(n) 2x log n 7=0 7+1 sin 1/2 where kn(t) is Harmonic Summability Kernel for Fourier series. Proof. I k (t) I = -2-i-;- 1 A 1 sin (n - y + l/2)t I 2t log n y=o y + 1 sin t/2 / 1 A ^_ (2n - 2y +1) sin t/2 \ \log w 7=o I sin 1/2 / \ log» 7=o 7+1/ = O(ra). Lemma 4. For < sracfe /Aa/ l/wg/^5.

3 1958] HARMONIC SUMMABILITY OF DOUBLE FOURIER SERIES 981 Proof. i*.wi =or-^ {i+iogi//}i. U log n J 1 «1 sin(n-y+l/2)t kn(t) = ;-Z-:-;- 2x log n y=o 7+1 sm t/2 = -:-2Z -»sin (n-\-) t cos (7 + 1)/ 2ir log n sin t/2 T_ I \ 2 / cos ( n H-11 sin (7 + 1)/ > 1 I / 3 \» cos (7 + 1)/ = -\ sin I n - -I / x log «sin t/2 \ \ 2 / 7=0 7+1 \w\=o\^\±c^±^\+ / 3 \» sin (7 + l)n -COs[n+-y^ -Tr-y ±^M\] Ltlogw I. 7_0 7+1 I 7=0 7+1 / J = o\--ll + log 1. by Lemma 1 and 2 Lemma 5. For t such that h^trzlir I MO I = o\-± L / log n 1. J Proof. Applying Abel's transformation, we have \kn(t)\ -ol-t W-t-T--^) L /log n 7_0 \ / sin [(«+ 1/2) + v - 1/2]/ sin»//2 "1 [" 1 "I sin t/2 J L nt log w J L / log w J 3. Proof of the theorem

4 982 P. L. SHARMA [December /IT I o Jo» X </>(«, v)km(u)kn(v)dudv I + I I + I I + I )*(«, v)km(u)kn(v)dudv 0 Jo Jo Jr Jfi Jo Jd Jr / = h h + h say. where l/m<5<ir, l/n<t<ir I /* = 0 ( - ' dudv ) \ log m log «J s J r uv / = o(l) by the help of Lemma 1. Iz = \ km(u)du I <p(u, v)kn(u)dv J 5 / 0 /> TT f% Xjn /* T (%T km(u)du I 4>(u, v)kn(v)dv + I km(u)du I #(m, v)kn(u)dv = /3,i + is,2 say. By Lemmas 3, 4 and the theorem, i i r n clln\ i i I 2s,i =0- I 4>(u, v)\dv\ L log m J o J /«.» =o\-- f \<b(u,v)\ -fl + log \dv~\ L log m J i/ v log n \ v / J T 1 PT i i*l = 0- <b(u, v) L log w log w J i/» J +o - - r i *<«, v) i log dv\ L log ra log» J i/n v v J = 23,2, ,2,21 say. 1 r 1 "1T 1 rr dv~ 23,2,1 =0-;- $(M,») +-- I $(«,») _logralog»l» Ji/ log w log w J i/b ir_ = 0r ' n +.r ' f'-^-l _ log m log wj L log m log njx/n v log l/»j,<s, rlo log ^v = o(l) + o\ - L log n J i/

5 1958] HARMONIC SUMMABILITY OF DOUBLE FOURIER SERIES 983 T 1 TT l I!! 1 I 23,2,21=0 --;- I I <i>(u, v) log dv\ L log ra log n J X/n v v J r i r i it = 0-3>(m, v) log LlogralogwL v v J i/ 1 rt ( 11 dv\ >(u, v)h +log-\-\ log m log n J i/ 1 v) v'j / 1 rt *(M> v) \ = o(y) + 0[ - -^-^-dv) \logralog n J i/ d2 / / 1 rt $(«, d) 1 \ + 0( - I -^-^log d»j \logra log n J i/ u2 v / = o(-±-ft -J^-\ + 0(J^r t\ \\0gn J Un nlogl/i)/ \ log W J l/n D/ / r log log 1/iHr \ ; /flog 1/py \ \L log n Alln) \L log n J l/n/ Thus i3 =o(l). Let us evaluate Similarly ^21 =o(l). Ix. a 1 l/m *% l/n n 1/m n r n 8 p l/n a j /* T \ 0 J Jo +J JO J l/n +J J l/m J 0 +J J l/m J l/n/ ) = ii.i + 71>8 + 7i,3 + 7i,4, say. = «(D. <b(u, v)km(u)kn(v)dudv a l/m /» l/n \ I < («, t>) I &m(w) kn(v) <2w<2» 1 a l/m <» l/n \ I <^)(m, j;) mndudv 1 Ul/m. r -i radw I </>(«, v)kn(v)dv = 0 I < (w, v)kn(v)dv L / l/n = o(l) as 23,2.

6 984 P. L. SHARMA [December Thus 7"i 2 =o(l). Similarly 7ii8 =o(l). I\a = I I 4>(u, v)km(u)k (v)dudv. J l/m J I In \li.i\=o\f f \<j>(u,v)\--ji + log-1-- LJi/mJi/n ulogm \ u)v\ogn r rs ct i i l l i = 0 I I I <b(u, v) -dudv LJi/mJi/n ulogm v log n J \l + log > dudv r f5 CT i i * l >(u, v) -log-dudv LJi/mJi/n ulogm u v log n J + 0 If { < («, v) I -log dudv LJiimJi/n ulogm v log n v J r r8 rri 1 * * * I I 0(«,») I -log-log dudv LJi/mJi/n ulogm u v log n v J = 7i,4,i I + 7l,4,2 + I 774,3 I + I 7i,4,4 I. 1 1 r cs l ct \*(#'*)I I Ii,i,\ 1=0 I - du I -dv LJi/mU log m J un v logn = 0\ r l*(u'v)l dv] L log n J 1/n v J = o(l) by the help of 73,2,i. [/" 1 rs "I I /i,4,21=0 I -dv I I <b(u, v) I -log du \_J i/n v log n J i/m u log m u J r 1 rs,, l 1 " = 0 - I I <b(u, v) \ log du L log m J l/m u u A I In,z\ = o(l) by h,t,t- = o(l) as I /1.4.2jr rs rt1 1 1 l 1 * T/i.4.4 =0 I I <*>(«,») I -log-log dudv. L J i/m J i/n ulogm u v log n v J By partial integration for double integral we have [2],

7 1958] HARMONIC SUMMABILITY OF DOUBLE FOURIER SERIES Ho, r) -log -log 5 log ra 5 5 log n r log l/u -log f 4>(m, r)-du t log n r J i/m u2 log ra 1 1 rt 1 log 1/s -.-logy *(«,») 7j-* 5 log ra 5 J i/n t> log w, f' f7^/, (1 - log 1/«)(1 - log 1/b) + I I $(m,!)) - flmfll) J i/m J i/n wv log ra log w = Lx + L Li. Lx a,s i \ / rr log i/m \ 4>(m, t)-du) + ol I < >(m, t)-<2m ) l/m M2 log ra / \Jl/, M2 / = o(l) by 73,2,i and 73,2,2-2,3 = o(l) as in 72. /'5 r T *(w, v)dudv i/m J i/n uvlogmlogn /'8 rt $(u,v) log (l/u)dudv J'6 /'5 l/m / l/n «V log ra log W i/m J i/n i/m J i/n rt < («,») log (\/v)dudv uvlogmlogn rt $(w,») log (I/m) log (l/v)dudv uvlogmlogn = 74,1 + 74,2 + 74,3 + 74,4. z,i = 0(fs-^r-^\ \ J l/m M log 1/M J l/n n log 1/d/ a'6 /" 1 1 l \ i/m J i/n u v log 1/h log ra log w / a'5 <2«/*t (2m\ = 0(1). i/m u log ra J l/ u log l/»/

8 986 P. L. SHARMA Similarly 4,3 = 0(1). a,s du rr dv Um ulogm J i/n v log n Thus the proof is complete. I am much indebted to Dr. M. L. Misra for his kind help and guidance in the preparation of this paper. References 1. Y. S. Chow, On the cesaro summability of double Fourier series, Tohoku Math. J. vol. 5 (1953) pp J. J. Gergen, Convergence criteria for double Fourier series, Trans. Amer. Math. Soc. vol. 35 (1933) pp Hardy and Rogosinki, Proc. Cambridge Philos. Soc. vol. 43 (1947) pp J. G. Herriot, The Nbrlund summability of double Fourier series, Trans. Amer. Math. Soc. vol. 52 (1942) pp Hille and Tamarkin, On the summability of Fourier series, Trans. Amer. Math. Soc. vol. 34 (1932) pp B. N. Prasad and Siddiqui, Harmonic summability of Fourier series, Proc. Indian Acad. Sci. vol. 28 (1948) pp E. C. Titchmarsh, Theory of functions, p. 40. University of Saugar, Sagar, India

ON THE GIBBS PHENOMENON FOR HARMONIC MEANS FU CHENG HSIANG. 1. Let a sequence of functions {fn(x)} converge to a function/(se)

ON THE GIBBS PHENOMENON FOR HARMONIC MEANS FU CHENG HSIANG 1. Let a sequence of functions {fn(x)} converge to a function/(se) for x0