Approximation of functions of several variables by linear methods in the space Sp

Abstract

In the spaces S p of functions of several variables, 2π-periodic in each variable, we study the approximative properties of operators A △ ̺,r and P △ ̺,s , which generate two summation methods of multiple Fourier series on triangular regions. In particular, in the terms of approximation estimates of these operators, we give a constructive description of classes of functions, whose generalized derivatives belong to the classes S p H ω .Keywords and phrases: space S p , classes H ω , linear methods.Mathematics subject classification: 42B05, 26B30, 26B35.

Approximation theorems for multivariate Taylor-Abel-Poisson means

“…The operators A ̺,r were first studied in [15] where, in the terms of these operators, the author gave the structural characteristic of Hardy-Lipschitz classes H r p Lip α of functions of one variable, holomorphic on the unit disc of the complex plane. In [17], in terms of approximation estimates of such operators in some spaces S p of Sobolev type, the authors give a constructive description of classes of functions of several variables whose generalized derivatives belong to the classes S p H ω . In [13], direct and inverse approximation theorems of 2π-periodic functions by the operators A ̺,r were given in the terms of K-functionals of functions generated by their radial derivatives.…”

Approximation of functions by linear summation methods in the Orlicz-type spaces

“…The operators A ϱ,r were first studied in [15], where the author gave a structural characteristic of the Hardy-Lipschitz classes H r p Lip α of one-variable functions, holomorphic in a unit disk on the complex plane in the terms of these operators. Approximative properties of these operators were also considered in [13,16]. In the general case, the operators P ϱ,s were, perhaps, first considered as the aggregates of approximation of functions of one variable in [3,4].…”

“…In S p , the approximative properties of linear summation methods of Fourier series were studied in [16,17]. The purpose of this paper is to continue this study of the approximative properties of linear summation methods in the spaces S M .…”

Exact constants in direct and inverse approximation theorems for functions of several variables in the spaces Sp

“…In the spaces S p , the approximative properties of linear summation methods of multiple Fourier series on triangular regions were studied in [12,13]. In particular, in [13], the authors proved direct and inverse theorems of approximation of function of several variables by the Taylor-Abel-Poisson operators in S p .…”