Approximation on hexagonal domains by Taylor-Abel-Poisson means

TL;DR

This paper investigates the approximation capabilities of Taylor-Abel-Poisson means for multivariable periodic functions on hexagonal domains, establishing key theorems and inequalities in the context of Fourier series summation.

Contribution

It introduces new approximation theorems and Bernstein inequalities for Taylor-Abel-Poisson means on hexagonal domains, advancing understanding of Fourier series summation in this setting.

Findings

Proved direct and inverse approximation theorems.

Established Bernstein-type inequalities for radial derivatives.

Analyzed approximation properties in the integral metric.

Abstract

Approximative properties of the Taylor-Abel-Poisson linear summation me\-thod of Fourier series are considered for functions of several variables, periodic with respect to the hexagonal domain, in the integral metric. In particular, direct and inverse theorems are proved in terms of approximations of functions by the Taylor-Abel-Poisson means and $K$-functionals generated by radial derivatives. Bernstein type inequalities for $L_1$-norm of high-order radial derivatives of the Poisson kernel are also obtained.