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

Year 1981, Volume: 30 , 0 - 0, 01.01.1981

Z.u. Ahmad

https://doi.org/10.1501/Commua1_0000000094

Abstract

For the class of Fourier serîes with 8-quasîmonotone coefficients, itiş proved tbat I i Sn-'Jn I I = 0(0. “ co , if and only if a^^ log n = o(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
f(x) =
co + 2
n=ı
12 »n cos nx
has been settied for various special class of coefficients, (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 ([6], Theorem 1).

Keywords

L-Convergence, Fourier series, s-quasi monotone

References

• Ankara Üniversitesi – Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Dergisi