On approximation of functions, conjugate to the functions of several variables belonging to weighted Lipschitz and Zygmund classes

Year 2024, Volume: 53 Issue: 6, 1497 - 1514, 28.12.2024

Shailesh Kumar Srivastava , Sachin Devaiya

https://doi.org/10.15672/hujms.1318033

Abstract

In the present paper, we introduce a new weighted Lipschitz class $W(L^p(T^N),\xi_1(s_1),\dots,\xi_N(s_N))$ and Zygmund class $Z(L^p(T^N),\xi_1(s_1),\dots,\xi_N(s_N))$ for $N\in\mathbb{N}$, which generalizes the classes given in [12, 16]. We prove two theorems about the degree (error) of approximation of functions, conjugate to the $N$-variable functions ($2\pi$-periodic in each variable) belonging to these classes using the $N$-multiple matrix means of their $N$-multiple conjugate Fourier series. We improve the results of Móricz and Rhoades [11] and M\'{o}ricz and Shi [12], which are given in the form of corollaries.

Keywords

Degree (error) of approximation, weighted Lipschitz class, weighted Zygmund class, N-multiple matrix means

Project Number

Award No.: 09/1007(0008)/2020-EMR-I; Grant No. 2020- 21/Seed Money/26

Thanks

This work was supported by the Council of Scientific and Industrial Research (CSIR), New Delhi, India [Award No.: 09/1007(0008)/2020-EMR-I], and Sardar Vallabhbhai National Institute of Technology, Surat-395007, Gujarat [Grant No. 2020- 21/Seed Money/26].

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