Approximation by matrix transforms in weighted Orlicz spaces

Sadulla Z Jafarov

doi:10.3906/mat-1909-44

Abstract

In this work the approximation problems of the functions by matrix transforms in weighted Orlicz spaces with Muckenhoupt weights are studied.We obtain the degree of approximation of functions belonging to Lipschitz class $Lip(\alpha ,M,\omega )$ through matrix transforms $% T_{n}^{\left( A\right) }(x,f)$,$~$and Nörlund means $N_{n}\left(x,f\right) ~$of their trigonometric Fourier series.

Highlights

Let T denote the interval [−π, π], C the complex plane, and Lp(T), 1 ≤ p ≤ ∞, the Lebesgue space of measurable complex-valued functions on T
We will say that M satisfies the ∆2− condition if M (2u) ≤ cM (u) for any u ≥ u0 ≥ 0 with some constant c, independent of u
We investigate the degree of approximation of functions belonging to Lipschitz class Lip(α, M, ω) through matrix transforms Tn(A)(x, f ) and Nörlund means Nn (·, f ) of their trigonometric Fourier series

Summary

Introduction

Let T denote the interval [−π, π] , C the complex plane, and Lp(T) , 1 ≤ p ≤ ∞ , the Lebesgue space of measurable complex-valued functions on T. Let N be the complementary Young function of M It is known that the indices αM and βM satisfy 0 ≤ αM ≤ βM ≤ 1, αN + βM = 1 , αM + βN = 1 and the space LM (T) is reflexive if and only if 0 < αM ≤ βM < 1. Let us indicate by Ap (T) the set of all weight functions satisfying Muckenhoupt’s Ap -condition on T. According to [35], [36, Lemma 3.3], and [36, Section 2.3], if LM (T) is reflexive and the weighted function ω satisfies the condition ω ∈ A1/αM (T) ∩ A1/βM (T) , the space LM (T, ω) is reflexive.

Nörlund means

Pn n

Now we estimate n

Taking into account