Abstract
We study the Walsh series expansion of multivariate functions in Lp (1≤p≤∞) and, in particular, in Lip(α,p). The rate of uniform approximation by T-transformation of rectangular partial sums of double Walsh to these functions is investigated. By extending the concepts of rest (head) bounded variation series, which was introduced by Leindler (2004), we generalize the related results of Móricz and Rhoades (1996), Nagy (2012). Our results can be applied to many summability methods, including the Nörlund summability and weighted summability.
Highlights
Let r0(x) be the function defined on I := I0 = [0, 1) by{{{1, r0 (x) = {{{{−1, x x ∈ ∈ [0, ) [ 1, 1) r0 (x + 1) =
Moricz and Siddiqi [11] studied the rate of uniform approximation by Norlund means of Walsh (Walsh-Fourier) series of f ∈ Lp[0, 1)
Nagy [13] did some research on the approximation by Norlund means of double Walsh-Fourier series for Lipschitz functions and generalized Theorems A and B to the functions of two variables
Summary
Moricz and Siddiqi [11] studied the rate of uniform approximation by Norlund means of Walsh (Walsh-Fourier) series of f ∈ Lp[0, 1). Moricz and Rhoades [12] studied the corresponding approximation problem by weighted means of Walsh-Fourier series. Their main results in [12] can be read as follows. Nagy [13] did some research on the approximation by Norlund means of double Walsh-Fourier series for Lipschitz functions and generalized Theorems A and B to the functions of two variables. ∞; let {qjk} be a double sequence of nonnegative numbers such that it is nondecreasing; Δ 11qjk is of fixed sign and satisfies the regularity condition:. We will see that Theorems A, B, and C are corollaries of our results and some other new types of estimates are presented in this paper
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