Iterat. J. Math. & Math. Sci. VOL. 12 NO. (1989) 99106 99 ON (J,p) SUMMABILITY OF FOURIER SERIES S.M. MAZHAR Departmet of Mathematics Kuwait Uiversity P.O. BOX 5969 13060, Safat, Kuwait (Received December 13, 1986 ad i revised form April 8, 1987) ABSTRACT. I this ote two theorems have bee established. The first oe deals with the summabillty (J,p) of a Fourier series while the secod o cocers with the summabillty of the first derived Fourier series. These results iclude, as a special case, certai results of Nada [I]. KEY WORDS AND PHRASES. (J,p) summsbility, Fourier series, Derived Fourier series. 1980 AMS SUBJECT CLASSIFICATION CODES. 42A24, 40GI0. INTRODUCTION. Let {P be a sequece of oegatlve umbers such the radius of covergece of power series o P diverges ad let pcz) p x (1.1) be I. Give ay series I a with the sequece of partial sums {s we write ad Ps(X) E PS x (1.2) PS (x) J (x)= (I 3) If the series i (1.2) is coverget i [0,I) ad lira J (x) s, s x+l we say the series E a or the sequece {s is summable (J,p) is a fiite umber. ([2], [3], p.80). For P=l, with to s, where s Po=0, ad A k, k > I we get summablllty A, summabllity (L) ad A method of summability respectively. k
i00 S.M. MAZHAR Suppose f is a periodic fuctio with period 2 ad itegrable i the sese of Lebesgue over (,). ~ Let f(x) + Z (a cos x + b si x) E A(X). (1.4) The the first derived series of (1.4) is r. (b cos x a si x) z B (x). (1.5) We write @(t) = {f(x 0 + t) + f(x 0 t) 2s} *(t) = {f(x 0 + t) f(x 0 t)} g(t) (t) 2si t/2 s M(t) o Z P x si t ad (t) f *(u.):u du 2. MAIN RESULTS. (t) g(u) u du. (J,p) I this ote we propose to establish the followig theorems o summabillty of (1.4) ad (1.5). THEOREM I. Let {p be a positive sequece such Pk (a) P 0(I), r. 0(pv ad (b) E A2( p. x)l 0(lx), 0 < x < I. O If #(t) o(p(lt)), t +0. the the Fourier series (1.4) is summable (J,p) THEOREM 2. Let {p satisfy the hypothesis (a) of Theorem I. If to s. (c) oz )A ( P x)l O(lx), 0 < x < ad G(t) o(p(lt)), as t+ 0+, the the first derived series (1.5) is summable (J,p) to s. It may be remarked for P ) I, Po 0 our theorems iclude two kow theorems of Nada [I] o Lsummabillty of Fourier series ad its derived series. For a earlier result o (J,p) summabillty of (1.4) uder more striget coditios see
SUMMABILITY OF FOURIER SERIES I01 Kha [4]. Very recetly i 1985 Prem Chadra, Mohapatra ad Sahey [5] have established a similar theorem o (J,p) summabillty with aother set of coditios. 0(I) p (x) 0(x) It may be observed as x I Also it is easy to P see P < (+l) P+l 0,1,2,... ad p (x) 0( imply 0(I). P For if { p is ot bouded the llm. Now usig the well kow result P + 0 I i c x/l 7. x o where radius of covergece of each power series is ad a > 0 with lim (lx) P x. This 7., we fid for e I, 8 o P meas llm (I x)p (x) which cotradicts the ypothesis (lx)p (x) 0(I) as /I Thus coditios (2.1) ad (2.4) of [5] imply P 0(I). 3. PROOF OF THEOREM i. so Let s (x O) deote the th partial sum of (1.4) at x x O. 2 si t _f0 (t) at + o(i) t E0 P x(s(x0) s) f0 =E0 P s (x O) s 2 :f0 2 @ (t)m(t) dt + o(p(x)) /0 (t) E p x cos t dt + o(p(x)) o 2 lx =(f0 +fx. )... + o(p(x)) The 2 x si t dt + o(p(x)) I + 12 + o(p(x)), say. (3.1) Now 2 lx (t) x Ii = f0 =E0 P cos t dt lx ok f0 o(p(lt)) at] f x
102 S.M. MAZHAR Kha [4]. Very recetly i 1985 Prem Chadra, Mohapatra ad Sahey [5] have summabillty with aother set of coditios. established a similar theorem o (J,p) It may be observed 0(I) p (x) P 0(T_x) see P < (+l) P+l 0,1,2,... ad p (x) 0( ẋ) imply 0(I). P For if I p is ot bouded the llm P " + as x I Also it is easy to Now usig the well kow result I r. o x i x+l I ax o where radius of covergece of each power series is ad a > 0 with lim (lx) P x. x/l This Z a, we fid for a I, 6 P o meas lim (I x)p (x) which cotradicts the x+lhypothesis (lx)p (x) 0(I) as x I Thus coditios (2.1) ad (2.4) of [5] imply P 0(I). 3. PROOF OF THEOREM I. so Let s (x 0) deote the th partial sum of (1.4) at x x 0. 2 si _t dt + o(1) s (x 0) s fo (t) t Zo P x(s(x0)s) f0 2_ f @ (t)m(t) dt + o(p(x)) x 0 =2 fo (t) Z p o =2 fx/ fxx) " / o(pcx)) I + 12 t ZO P The 2 (t) x si t at + o(p(x)) x cos t dt + o(p(x)) + o(p(x)), say. (3.) Now 2 1x II = fo (t) ZO px cos t dt o( x (3.2) l"x k o( (lt) at lx fo ffieo Pk
SUMMABILITY OF FOURIER SERIES 103 o(1) Pk _xk+l Pk k kzo ky ( o()k.z0 k+t Z0 x x v Pk o(i) v o(p(x) Z0 k,zv o(i) 0 px Thus Agai where I o(p(x)). 2 k 12 = flx *(t) k.0 k x cos kt Pk! " z xk) Fk(t 7 lx (t) k.z0 (k Pk k k k sl(v)t Fk(t) v0 D(t) Z0 Dv(t) Z0 2 si t/2 (3.3) Uder the hypothesis of Theorem (1 x) 12 0(I) flx p(1t) J, dt t I (lx) dt o(p(x)) lx t 2 o(p(x) (3.4) Thus i view of (3.1), (3.3) ad (3.4) Z o P (S(X0) s) o(p(x)) as x I x This proves Theorem 1. 4. PROOF OF THEOREM 2. As show i ([6], p. 54) we ca assume s 0. so Let T(X0) deote the th partial sum of (1.5) at x x 0. T(X 0) = 0 f0 g(t) si t 2 0 si t/2 dt_f0 cos ( + ) The t g(t) dt 2 g(t) si t 2 w f dt + o(i) cos ( t +) t g(t) dt T + o(1) + T say,1, 2 Z T(X0) P x Z Tl P x + o(p(x)) + O O T2 P x L + o(p(x)) + L 2 say. (4.1)
104 S.M. MAZHAR As show i the proof of Theorem i view of (4.5), L o(p(x)). (4.2) cos( + )t g(t) dt g(t) 0 Z0 P x cos( +) t dt t G (t)h(t) dt d (th(t))}dt _2 f G(t) R(t) + th (t)} dt + say. L2 _2 Eo p x /0 2 f 2 2 fo g(t) H(t) dt fo _2, {JiG(t) li(t)] 0 0 G(t) 0 2 lx fo + Let H() z p x cos( + ) t. The Now sice P 0(I) w /Ix L21 L22 o(p(lt) L21 x l:x dt + fx o(p(lt)) t E dt o o(p(x) + o( J, dt o(p(x) 1x (4.3) as show i (3.2). I view of the hypothesis of Theorem 2. Sice 0(I), we have by usig Abel s trasformatio P d d d t H(t) g P x t cos( +) t Z a( p x) d t (si (+l)t) o 2 si t/2 E A2( x d { t o P 4 sl 2 t/2 Thus 2 a d L22 flx o(p(1t) [ (t H(t))l dt (4.4) (cos t/2 cos( + 3/2)t)}.
SUMMABILITY OF FOURIER SERIES 105 t 2 t 2 t 2 where C is a positive costat ot ecessarily the same at each occurrece, ad i view of the fact o la3( P x) O(lx) (4.5) implies )82( P x)) O(l"X) 0 Hece from (4.4) (lx) o(1) lx p(1t) dt o(p(x)) as show i (3.3). (4.6) t2 Thus from (4.1) (4.3) ad (4.6) the proof of Theorem 2 follows. REFERENCES I. NANDA, M. The summability (L) of the Fourier series ad the first differetiated Fourier series, Qu_a.rt. J. Math. Oxford (2) 13 (1962), 229234. 2. BORWEIN, D. O methods of summability based o power series, Proc. Royal Soc. Ediburgh 64 (1957), 342349. 3. HARDY, G.H. Diverget Series (Oxford, 1949). 4. KAN, F.M. O (J,p) 18 (II) (i972), 1317. summabillty of Fourier series, Proc. Ediburgh Math. Soc. 5. CHANDRA, P., MOHAPATRA, R.N. ad SAHNEY, B.N. Tamkag J. Math. 16 (1985) No. 2, 3741. 6. MOHANTY, R. ad NANDA, M. The summabtltty by logarithmic meas of the derived Fourier series, _Quart. J. Math. Oxford (2) 6 (1955), 5358.
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