1. Introduction. g(x) = a 2 + a k cos kx (1.1) g(x) = lim. S n (x).

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1 Georgia Mathematical Joural Volume 11 (2004, Number 1, INTEGRABILITY AND L 1 -CONVERGENCE OF MODIFIED SINE SUMS KULWINDER KAUR, S. S. BHATIA, AND BABU RAM Abstract. New modified sie sums are itroduced ad a criterio for the L 1 -covergece of these modified sie sums uder a ew class K is obtaied. Also a ecessary ad sufficiet coditio for the L 1 -covergece of the cosie series is deduced as a corollary Mathematics Subject Classificatio: 42A20, 42A32. Key words ad phrases: L 1 -covergece, modified sie sums, cojugate Fejer erel. Cosider the cosie series 1. Itroductio g(x = a a cos x (1.1 with partial sums defied by S (x = a a cos x ad let g(x = lim S (x. Cocerig the L 1 -covergece of cosie series (1.1 Kolmogorov [2] proved the followig theorem: Theorem A. If {a } is a quasi-covex ull sequece, the for the L 1 - covergece of the cosie series (1.1 it is ecessary ad sufficiet that lim a log = 0. The case of this theorem i which the sequece {a } is covex, was established by Youg [7]. That is why this Theorem A is sometimes ow as Youg Kolmogorov Theorem. Rees ad Staojević [5] have itroduced modified cosie sums g (x = 1 2 a + =0 j= ( a j cos x. Garret ad Staojević [1], Ram [4] ad Sigh ad Sharma [6] studied the L 1 -covergece of these cosie sums uder differet sets of coditios o the coefficiets a. ISSN X / $8.00 / c Helderma Verlag

2 100 KULWINDER KAUR, S. S. BHATIA, AND BABU RAM Later o, Kumari ad Ram [3], itroduced ew modified cosie ad sie sums as f (x = a ( aj cos x j ad g (x = j= j= ( aj j si x ad have studied their L 1 - covergece uder the coditio that the coefficiets a belog to differet classes of sequeces. Also, they deduced some results about the L 1 -covergece of cosie ad sie series as corollaries. We itroduce here ew modified sie sums as K (x = 1 j= ( a j 1 a j+1 si x. (1.2 The aim of this paper is to study the L 1 -covergece of these modified sie sums K (x ad to obtai a aalogue of Theorem A of Kolmogorov for a ewly defied class K of coefficiet sequeces defied as follows: Defiitio. If a = o(1,, ad 2 a 1 2 a +1 < (a0 = 0, (1.3 the we say that {a } belogs to the class K. 2. Mai Result The mai result is the followig theorem: Theorem 1. Let the sequece {a } belog to the class K, the K (x coverges to g(x i L 1 -orm. Proof. We have S (x = a = 1 = 1 a cos x = 1 a cos x a [si( + 1x si( 1x] si x (a 1 a +1 si x + a +1 + a si( + 1x.

3 INTEGRABILITY AND L 1 -CONVERGENCE 101 Applyig Abel s trasformatio, we have S (x = 1 ( ( a 1 a +1 D (x + (a a +2 D (x si x + a +1 + a si( + 1x where D (x deotes Dirichlet cojugate erel. Thus g(x = lim S (x = 1 ( a 1 a +1 D (x, if the series is coverget. Also, K (x = 1 ( a j 1 a j+1 si x j= = 1 ( (a 1 a +1 si x (a a +2 D (x. Applyig Abel s trasformatio, we have K (x = 1 ( a 1 a +1 D (x. Sice D (x = O( ad 2 a 1 2 a +1 <, the series 1 ( a 1 a +1 D (x coverges. Hece lim K (x = g(x exists. Thus g(x K (x = 1 =+1 ( 1 = lim m, ( a 1 a +1 D (x m =+1 ( a 1 a +1 D (x. The use of Abel s trasformatio gives g(x K (x = 1 [ m 1 lim ( + 1 ( 2 a 1 2 a +1 F (x m =+1

4 102 KULWINDER KAUR, S. S. BHATIA, AND BABU RAM ] +(m+1 ( a m 1 a m+1 F m (x ( + 1 ( a a +2 F (x = 1 [ ( + 1 ( 2 a 1 2 a +1 F (x =+1 ] ( + 1 ( a a +2 F (x, where F (x = deotes the cojugate Fejer erel. Now g(x K (x dx j=0 D j (x [ = 1 ( + 1 ( 2 a 1 2 a +1 F (x =+1 ( + 1 ( a a +2 F (xdx [ C =+1 ( + 1 ( π 2 a 1 2 a +1 + ( + 1 ( a a +2 F (x dx ]. F (x dx But 1 π F (x dx = 1, ad ( ( a a +2 = 2 a 2 a +2 = = = ( 2 a 1 2 a +1 =+1 ( ( 2 a 1 2 a +1 1 = o. + 1 Thus we have ( g(x K (x dx = O ( + 1 ( 2 a 1 2 a +1 =+1

5 INTEGRABILITY AND L 1 -CONVERGENCE o(1 = o(1 by (1.3. Corollary. If {a } belogs to the class K, the the ecessary ad sufficiet coditio for the L 1 -covergece of the cosie series (1.1 is lim a log = 0. Proof. We have S (x g(x S (x K (x + K (x g(x = K (x g(x Also, ad Sice + (a a +2 D (x + a si x +1 + a si( + 1x (a a +2 D (x + a si x +1 + a si( + 1x Moreover, = K (x S (x K (x g(x + S (x g(x, (a a +2 = ( a a +2 = = =. ( a 1 a +2 a +1 + a +2 =+1 D (x dx = O(, we obtai (a a +2 a si x +1 + a si( + 1x dx a si x ( ( 2 a 1 2 a +1 1 = o. D (x dx = O( (a a +2 = o(1. + si( + 1x dx = a Sice K (x g(x = o(1 (, by Theorem 2.1. D (x dx (a log.

6 104 KULWINDER KAUR, S. S. BHATIA, AND BABU RAM Therefore lim if ad oly if lim a log = 0. g(x S (x dx = o(1 Refereces 1. J. W. Garrett ad Č. V. Staojević, O L1 -covergece of certai cosie sums. Proc. Amer. Math. Soc. 54(1976, A. N. Kolmogorov, Sur l ordre de gradeur des coefficiets de la series de Fourier Lebesgue. Bull. Polo. Sci. Ser. Sci. Math. Astroom. Phys. 1923, Suresh Kumari ad Babu Ram, L 1 -covergece modified cosie sum. Idia J. pure appl. Math. 19(1977, No. 11, Babu Ram, Covergece of certai cosie sums i the metric space L. Proc. Amer. Math. Soc. 66(1977, C. S. Rees ad Č. V. Staojević, Necessary ad sufficiet coditio for itergrability of certai cosie sums. J. Math. Aal. Appl. 43(1973, N. Sigh ad K. M. Sharma, Covergece of certai cosie sums i the metric space L. Proc. Amer. Math. Soc. 75(1977, W. H. Youg, O the Fourier series of bouded fuctios. Proc. Lodo Math. Soc. 12(1913, No. 2, A. Zygmud, Trigoometric series. I. Cambridge Uiversity Press, Cambridge, Authors addresses: (Received Kulwider Kaur ad S. S. Bhatia School of Mathematics & Computer Applicatios, Thapar Istitute of Egieerig ad Techology (Deemed Uiversity P. B. No. 32, Patiala (Pb Idia address: math@hotmail.com Babu Ram Departmets of Mathematics Maharshi Dyaad Uiversity Rohta, Idia

A Note On L 1 -Convergence of the Sine and Cosine Trigonometric Series with Semi-Convex Coefficients

A Note On L 1 -Convergence of the Sine and Cosine Trigonometric Series with Semi-Convex Coefficients It. J. Ope Problems Comput. Sci. Math., Vol., No., Jue 009 A Note O L 1 -Covergece of the Sie ad Cosie Trigoometric Series with Semi-Covex Coefficiets Xhevat Z. Krasiqi Faculty of Educatio, Uiversity of