Rate of convergence of Fourier series on the classes of ¯ψ-integrals

Authors

  • O. I. Stepanets

Abstract

We introduce the notion of ¯ψ-integrals of 2π-periodic summable functions f, f ε L, on the basis of which the space L is decomposed into subsets (classes) L¯ψ. We obtain integral representations of deviations of the trigonometric polynomials Un(f;x;Λ) generated by a given Λ-method for summing the Fourier series of functions fεL¯ψ. On the basis of these representations, the rate of convergence of the Fourier series is studied for functions belonging to the sets L¯ψ in uniform and integral metrics. Within the framework of this approach, we find, in particular, asymptotic equalities for upper bounds of deviations of the Fourier sums on the sets L¯ψ, which give solutions of the Kolmogorov-Nikol'skii problem. We also obtain an analog of the well-known Lebesgue inequality.

Published

25.08.1997

Issue

Section

Research articles

How to Cite

Stepanets, O. I. “Rate of Convergence of Fourier Series on the Classes of ¯ψ-Integrals”. Ukrains’kyi Matematychnyi Zhurnal, vol. 49, no. 8, Aug. 1997, pp. 1069-13, https://umj.imath.kiev.ua/index.php/umj/article/view/5102.