Approximation by trigonometric polynomials in weighted Morrey spaces

Abstract

In this paper we investigate the best approximation by trigonometric polynomials in weighted Morrey spaces $\mathcal{M}_{p,\lambda}(I_{0},w)$, where the weight function $w$ is in the Muckenhoupt class $A_{p}(I_{0})$ with $1 < p < \infty$ and $I_{0}=[0, 2\pi]$. We prove the direct and inverse theorems of approximation by trigonometric polynomials in the spaces $\mathcal{\widetilde{M}}_{p,\lambda}(I_{0},w)$ the closure of $C^{\infty}(I_{0})$ in $\mathcal{M}_{p,\lambda}(I_{0},w)$. We give the characterization of $K-$functionals in terms of the modulus of smoothness and obtain the Bernstein type inequality for trigonometric polynomials in the spaces $\mathcal{M}_{p,\lambda}(I_{0},w)$.