On approximation in generalized Zygmund class

Hare Krishna Nigam

doi:10.1515/dema-2019-0022

Abstract

Abstract Here, we estimate the degree of approximation of a conjugate function g ˜ {\tilde g} and a derived conjugate function g ˜ ′ {\tilde g'} , of a 2π-periodic function g ∈ Z r λ 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.

Highlights

Here, we estimate the degree of approximation of a conjugate function gand a derived conjugate function g′, of a 2π-periodic function g ∈ Zrλ, r ≥ 1, using Hausdorff means of CFS and CDFS respectively
Some important corollaries are deduced from our main theorems
The study of error estimation of a conjugate function gof a 2π-periodic function g belonging to Lip(α), Lip(α, r), Lip(ξ (t), r) and W(Lr, ξ (t)) using single summability operators such as Euler, Cesàro, Nörlund, generalized Nörlund, Hölder, Karamata, Riesz, matrix and almost matrix, has been a centre of creative study for the several researchers like Kushwaha [1], Nigam and Sharma [2], Qureshi [3,4,5,6], Lal and Nigam [7], Lal

Summary

Introduction

The study of error estimation of a conjugate function gof a 2π-periodic function g belonging to Lip(α), Lip(α, r), Lip(ξ (t), r) and W(Lr , ξ (t)) using single summability operators such as Euler, Cesàro, Nörlund, generalized Nörlund, Hölder, Karamata, Riesz, matrix and almost matrix, has been a centre of creative study for the several researchers like Kushwaha [1], Nigam and Sharma [2], Qureshi [3,4,5,6], Lal and Nigam [7], Lal [8], Rhoades [9], Mittal et al [10], Mishra [11] and Kranz et al [12] in past few decades. It can be noted that the matrices involved in Cesàro, Hölder, Euler and their product are Hausdorff matrices1 Considering this view point, Singh and Srivastava [25] studied error estimates of a conjugate function gof a 2π-periodic function g ∈ W(Lr , ξ (t)) using Hausdorff means. Since the studies of error estimates of a conjugate derived function g′ of a 2π-periodic function g either in the Lipschitz space or in the Zygmund space have not been initiated so far. The modulus of continuity of second order satisfies conditions (i)-(iii) and a further condition, given as follows:. Readers may refer to the paper of Weiss and Zygmund [39] which dealt with conditions on the second order modulus of smoothness sufficient to force absolute continuity of a function.

Lemmas

Proof of Theorem 1

Proof of Theorem 2

Corollaries

Particular cases