Approximation of functions in generalized Zygmund class by double Hausdorff matrix

H K Nigam,Supriya Rani,M Mursaleen

doi:10.1186/s13662-020-02711-z

Abstract

In the present work, we emphasize, for the first time, the error estimation of a two-variable function g(y,z) in the generalized Zygmund class Y_{r}^{(xi )} (rgeq 1) using the double Hausdorff matrix means of its double Fourier series. In fact, in this work, we establish two theorems on error estimation of a two-variable function of g in the generalized Zygmund class.

Highlights

1 Introduction The study of the error estimation of a function of single variable g in Lipschitz spaces, Hölder spaces, generalized Hölder spaces, Besov spaces, Zygmund spaces, and generalized Zygmund spaces with different single means, and various product summability means of Fourier series and conjugate Fourier series have been considered as a center of creative study for the researchers [1,2,3,4,5,6,7,8,9,10,11,12,13] in the past few decades
The above review of research shows that the studies of error estimation of a twovariable function g(y, z) in the generalized Zygmund class Yr(ξ) (r ≥ 1) using double Hausdorff means of double Fourier series have not been initiated so far
The basic theory of Hausdorff transformations for double sequences came into being by Adams [16] in 1933

Summary

Introduction

The study of the error estimation of a function of single variable g in Lipschitz spaces, Hölder spaces, generalized Hölder spaces, Besov spaces, Zygmund spaces, and generalized Zygmund spaces with different single means, and various product summability means of Fourier series and conjugate Fourier series have been considered as a center of creative study for the researchers [1,2,3,4,5,6,7,8,9,10,11,12,13] in the past few decades. The above review of research shows that the studies of error estimation of a twovariable function g(y, z) in the generalized Zygmund class Yr(ξ) (r ≥ 1) using double Hausdorff means of double Fourier series have not been initiated so far. We consider the error estimation of two-variable functions g(y, z) in the generalized Zygmund class Yr(ξ) (r ≥ 1) by the double Hausdorff summability means of its double Fourier series. We establish two theorems on the degree of approximation of a two-variable function in the generalized Zygmund class Yr(ξ) (r ≥ 1) by the double Hausdorff summability means of its double Fourier series. Necessary and sufficient condition for double Hausdorff matrices to be conservative is the existence of a function χ(s, l) ∈ BV [0, 1] × [0, 1] such that dχ(s, l) < ∞.

Theorems

Lemmas

Conclusion