Approximation by Vilenkin-Fourier sums

Abstract

Introduction. In this paper we deal with the connection (in different spaces) among the Vilenkin--Fourier sums, the modulus of continuity and the Lebesgueconstants (with respect to the Vilenkin-system). We give two sided estimates for an expression containing these quantities. The corresponding problem for the trigonometric system was considered by Lebesgue [7] and Oskolkov [8].

Logarithmic means of Walsh-Fourier series

“…Uniform convergence of Walsh-Fourier series of the functions of classes of generalized bounded variation was investigated by author [10]. This problem has been considered for Vilenkin group by Fridli [3] and Gát [5]. Lukomskii [15] considered uniform and L 1 -convergence of subsequence of partial sums of Walsh-Fourier series.…”

A sharp boundedness result for restricted maximal operators of Vilenkin–Fourier series on martingale Hardy spaces

“…Then, in view of (8) and (9) we can conclude that By the definition of ̺ there exists at least one index k 0 ∈ N + such that ̺ = α k 0 . By using contition (5) we can conclude that…”

“…Uniform convergence of subsequences of partial sums was also investigated by Goginava and Tkebuchava [7]. This problem was considered for the Vilenkin group G m by Fridli [5], Blahota [3] and Gát [6].…”

Some inequalities related to strong convergence of Riesz logarithmic means

“…Remark 4. Approximation properties of some summability methods in the classical and real Hardy spaces were considered by Oswald [32], Kryakin and Trebels [18], Storoienko [30], [31] and for martingale Hardy spaces in Fridli, Manchanda and Siddiqi [9] (see also [7], [8]), Nagy [20], [21], [22], Tephnadze [39], [40], [41].…”