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 $\mathcal{M}_{p,\lambda}(I_{0},w)$.
Funding Statement
The research of Z. Cakir and C. Aykol was partially supported by the grant of Ankara University Scientific Research Project (BAP.17B0430003).
Citation
Z. Cakir. C. Aykol. D. Soylemez. A. Serbetci. "Approximation by trigonometric polynomials in weighted Morrey spaces." Tbilisi Math. J. 13 (1) 123 - 138, January 2020. https://doi.org/10.32513/tbilisi/1585015225
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