A note L-Convergence of Fourier series with s-quasi monotone Coefficients

Authors : Zu AHMAD

Pages : 0-0

Doi:10.1501/Commua1_0000000094

View : 14 | Download : 6

Publication Date : 1981-01-01

Article Type : Research Paper

Abstract :For the class of Fourier serîes with 8-quasîmonotone coefficients, itiş proved tbat I i Sn-`Jn I I = 0insert ignore into journalissuearticles values(0. “ co , if and only if a^^ log n = oinsert ignore into journalissuearticles values(l);, n CO . This generalizes the theorem of Garrett, Rees and Stanojevic [3], and Telyakovskii and Fomine [6] for quasi-monotone, and monotone coefficients respectively. 1. A seguence {a„} of positive numbers is said. to be quasi- monotone if Aa, —.a — for some positive k, where Aaj, »n —` ^n+ı. It is obvious tbat every null monotonic decreasing sequence is quasi-monotone. The sequeııce {aj,} is said to be S- quasi-monotone if a^‘n o, a.n o ultimately and Aa^ > — wbere {S^} is a sequence of positive numbers. Clearly a null quasi- monotone sequence is S-quasi-monotone witb Sn= n 2. The problem of L-*convergence of Fourier cosine seri es finsert ignore into journalissuearticles values(x); = co + 2 n=ı 12 »n cos nx has been settied for various special class of coefficients, insert ignore into journalissuearticles values(See e.g. Young [7], Kolmogorov [4], Fomine [1], Garrett and Stano­ jevic [2], Telyakovskii and Fomine [6], ete);. RecCntly, Garrett, Rees and Stanojevic [3] proved the fol­ îowing theorem which is too a generalization of a result of Telya- kovskii and Fcmine insert ignore into journalissuearticles values([6], Theorem 1);.
Keywords : L Convergence, Fourier series, s quasi monotone