(3) 6(t) = /(* /(* ~ t), (0 - /( + 0 ~ /(* ~t)-l,

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1 ON THE BEHAVIOR OF FOURIER COEFFICIENTS R. MOHANTY AND M. NANDA 1. Let/(0 be integrable L in ( ir,ir) periodic with period 2ir, let (1) f(t) ~ oo + E (a* cos ni + bn sin nt) = a0 + E ^n(0-2 i 2 i Then the conjugate series of (1) at t = x is (2) E (*»» cos nx an sin nx) = E -#«(*) i i We write (3) 6(t) = /(* /(* ~ t), *(0 - /(* + 0 ~ /(* ~t)-l, where I is a finite number. It is known1 that if ip(t) >0 as / >0, then the sequence jwb»(x)} is summable (C, r), r>l, to the value l/ir, but not necessarily summable (C, 1). It is also known2 that with the same hypothesis the sequence \nbn(x)} is summable by the first logarithmic mean to the value l/ir. The object of this note is to prove the following Theorem 1. If (4) Ht) = o((logj\ 1 as*^0 (5) an = 0(n~>), bn = 0(«-8), 0 < S < 1, then the sequence {nbn(x)} is summable (C, 1) to the value l/ir. 2. Proof of the theorem. We are to prove that 1 A / 2_, rbr(x) * asw > oo. n i it Writing tn(x) = E" ^Br(x), we have Received by the editors November 19, 1952, in revised form, June 30, See Zygmund, Trigonometrical series, 1935, p Zygmund, loc. cit. 79

2 80 R. MOHANTY AND M. NANDA (February where = - f '{/(* /(*- t)}g(n, t)dt, n t J o 1 n g(n, t) = E r sin r*. w i Denoting Jlg(n, t)dt by X, we have Therefore Thus 1 n X = E {1 - (-l)rl = 1 + ^(1), as»^«.» i 1 I 1 / * / (*)-X = {/(* /(x l\g(n, t)dt n IT 17 J 0 = f +(t)g(n, t)dt. IT J 0 1 I 1 rr Ux)-= I i(t)g(n, t)dt + o(l) (o) n ir it J o We have = 7 + o(l), say. Id.. g(», 0 =-{1/2 + cos I + cos 2* + + cos nt\ n dt Let 1 d Isin (n + 1/2)A n dt\ 2 sin (t/2) ) 1 sin (n + 1/2); cos (t/2) (n + 1/2) cos («+ 1/2)/ 4m sin2 (//2) 2» sin (</2) (7) i = r+(t)g(n,odt = r + r + ft=/i+/,+/i, J 0 / 0 / n-1 J n~» where r = S/2. We require the following inequalities (8) *(», 0 = O(n20, (9) *(«, 0 = 0(t->). for g(», /)

3 i954l on the behavior of fourier coefficients 81 Proof of (8). sin (n + 1/2)/ cos (t/2) - 2(n + 1/2) cos (n + 1/2)* sin (t/2) g(n, t) = -.,,.. 4» sin2 (t/2) By replacing the sines cosines in the numerator of the above fraction by their respective power series expansions it is easy to see that, for 0<2<1/m, the numerator is 0(nH) hence which completes the proof of (8). Proof of (9). Now Similarly Therefore, g(n, t) < K (nst3/nt2) = KnH, sin (n + 1/2)2 cos (t/2) (n + l/2)tir2 t2 4m sin2 (t/2) 4nt2 2 - (n + 1/2) cos (n + 1/2)2 - < ret l. 2w sin (t/2) which completes the proof of (9). Using (8) we find g(n, t) < Or + 7r2/2)ri, By (9) we have /» n~ / n o 4>(t)g(n, t)dt = I Jo o(\)0(n2t)dt = o(l). (11) \JA= \(-\ = o(l). 11 U - Mog (1/0/ Now we write (12) /, = /s.i - /3.2 where (13) 1 f* /3.i = I cos (t/2) sin (n+ 1/2)2 ^(2)-..,,,-*, 4«J n-r sin2 (2/2) ri^ r W+1/2 C a cos (M + W i (14) 7,.2 = - j /-(*)- -<fc. 2» J»-' sin (2/2)

4 82 R. MOHANTY AND M. NANDA [February We take up J3.2 first. Now the Fourier CO 2 E Bn(x) sin nt - I. i series for \p(t) is By multiplying the Fourier series of \p(t) by cos («+ l/2)//sin (t/2) which is of bounded variation integrating term by term, we have3 /cos (n + 1/2)/ " r ' 2 sin kt cos (n + 1/2)/ W)- dt = E s*- * - sin(//2) i J*- sin (//2) Now the coefficient of Bk equals / * cos (n + 1/2)/ - dt. /' * sin (n + 1/2 + k)t - sin (n + 1/2 - k)t -dt -" sin (//2) n-f sin (t/2) = (cosec (n-"/2)) I {sin (w + 1/2 + k)t - sin (n + 1/2 - *)/}<** J n~*. (n~r < < x) it follows that the coefficient of Bk does not exceed 47r«r/ («+ l/2) in absolute value. By the second mean value theorem, we find Thus, by (14), /» cos (n + 1/2)/ - ' ' dt = o(l). sin (t/2) Now i /,., < o(l) + 4xE k/n' = 0(1) + t+ t, k-i \ r (n + 1/2) * i *_ +i =. o(l) + Ri + i?2. <-E +»r ( ) E - 4x (n + l)/2 rfisi/i \ 2 /»/2StS» (» + 1/2) - A = 0(w-8/2) + 0(w-8'2 log n) = o(l), <»'«-5 E -7T +2»r E- 47T n<k^2n ~ («+ 1/2) it>2n A 1 _=_0(»-6'2 log n) + 0(«-{'2) = 0(1). 3 Zygmund, Trigonometrical series, The discussion of Js.t is effected by arguments similar to those in Zygmund has been given only for the sake of completeness.

5 igj4l ON THE BEHAVIOR OF FOURIER COEFFICIENTS 83 Hence (15) /,., = o(l). Again where 1 f " iko {sin (w + 1)2 + sin nt\dt ^' i = 77~ I -.,, s- = I» + It, 2m J -r sin2 (2/2) 1 r T sin (m + 1)2 (16) U = T) m»nm dl 2m J -r sin2 (2/2) 1 C T sin m2 (17) 74 = I *(/)-A. 2mJ - sin2 (2/2) By applying arguments similar to that used above it can be easily shown that Hence It-0(^-\-o(l). (18) /3.1 = o(l). Therefore by (12), (18), (15) we have (19) J3 = o(l). By (7), (10), (11), (19), we have (20) J = o(l). Therefore by (6) (20), 1 I 2 (x)-= o(l) as m > 00, n it this completes the proof of the theorem. 3. It is easy to see from Schwarz's example of a divergent Fourier series at a point of continuity slightly modified that condition (4)

6 84 R. MOHANTY AND M. NANDA alone does not imply summability (C, 1) of the sequence nb (x). Hardy-Littlewood's convergence criterion for the allied series is4 Theorem A. If A(i) 6(t) = o {(logy) } A(ii) each of an bn is 0(n~s), 0 < 5 < 1, then the allied series E Bn(x) converges to the value 1 rr (21) 0(/) cot (t/2)dt, 2irJ _o provided that the integral exists as a "Cauchy integral" at the origin. We shall now deduce Theorem A as a corollary of our theorem by employing the following known result also due to Hardy Littlewood.6 Theorem B. If EM» is summable (A), then a necessary sufficient condition that it should be summable (C, k), k) > 1, is that the sequence {nun} is summable (C, k-\-l) to the value 0. Proof of Theorem A. The existence of the integral (21) as a "Cauchy integral" at the origin implies the summability (A) of the allied series E" Bn(x).6 By using Theorem 1, we find that conditions A(i) A(ii) of Theorem A imply the summability (C, 1) of the sequence {«J3n(x)} to the value 0. Now the convergence of the allied series is a consequence of Theorem B. Ravenshaw College, Cuttack, India * Hardy Littlewood, Annali della R. Scuola Normale Superiori di Pisa (2) vol. 3 (1934) pp , Theorem 13. See also Zygmund, Trigonometrical series, 1935, p. 35, where the slightly stronger theorem, viz.: If (i) f(x+h) fix) =0 (log 1/ h\ )_1, (ii) each of a bn is 0(n~s), 5 >0, then Sn(x) fix, ir/n) =o(l) (in the notation of Zygmund), is given, which includes Theorem A. 8 Hardy Littlewood, J. London Math. Soc. vol. 6 (1931) p Zygmund, Trigonometrical series, 1935, p. 55.

Ht) = - {/(* + t)-f(x-t)}, e(t) = - I^-á«.

$Ht) = - {/(* + t)-f(x-t)}, e(t) = - I^-á«.$ ON THE ABSOLUTE SUMMABILITY OF THE CONJUGATE SERIES OF A FOURIER SERIES T. PATI 1.1. Let/( ) be a periodic function with period 2w and integrable (L) over ( ir, it). Let 1 (1.1.1) a0 + Y (an cos nt + bn