On approximation in generalized Zygmund class

Here, we estimate the degree of approximation of a conjugate function {\tilde g} and a derived conjugate function {\tilde g'} , of a 2π-periodic function g \in Z_r^\lambda , r ≥ 1, using Hausdorff means of CFS (conjugate Fourier series) and CDFS (conjugate derived Fourier series) respectively. Our main theorems generalize four previously known results. Some important corollaries are also deduced from our main theorems. We also partially review the earlier work of the authors in respect of order of the Euler-Hausdorff product method.

The degree of approximation by Hausdorff means of a conjugate Fourier series

The purpose of this paper is to analyze the degree of approximation of a function \(\overline f\) that is a conjugate of a function \(f\) belonging to the Lipschitz class by Hausdorff means of a conjugate series of the Fourier series.

On the degree of approximation of functions belonging to a Lipschitz class by Hausdorff means of its Fourier series

In a recent paper Lal and Yadav [1] obtained a theorem on the degree of approximation for a function belonging to a Lipschitz class using a triangular matrix transform of the Fourier series representation of the function. The matrix involved was the product of $ (C, 1) $, the Cesaro matrix of order one, with $ (E, 1) $, the Euler matrix of order one. In this paper we extend this result to a much wider class of Hausdorff matrices.

On the degree of approximation of functions belonging to the weighted $ (L^p, \xi (t)) $ class by Hausdorff means

In this paper we obtain a theorem on the degree of approximation of functions belonging to a certain weighted class, using any Hausdorff method with mass function possessing a derivative. This result is a substantial generalization of the theorem of Lal [2].

Lebesgue constants for double Hausdorff means

As is well known, the divergence of the set of constants known as the Lebesgue constants corresponding to a particular method of summability implies the existence of a continuous, periodic function whose Fourier series, summed by the method, diverges at a point, and of another such function the sums of whose Fourier series converge everywhere but not uniformly in the neighborhood of some point.In 1961, Lorch and Newman established that if L(n; g) is the nth Lebesgue constant for the Hausdorff summability method corresponding to the weight function g(u), thenwherewhere the summation is taken over the jump discontinuities {εk} of g(u) and M{f(u)} denotes the mean value of the almost periodic function f(u).In this paper, a partial extension of this result to the two dimensional analogue is obtained. This extension is summarized in Theorem 1.3.