Approximation of functions of several variables by trigonometric polynomials with harmonics from hyperbolic crosses
type: Item
instance of: scholarly article
title: Approximation of functions of several variables by trigonometric polynomials with harmonics from hyperbolic crosses
zbMATH Open document ID: 0714.42002
DOI: 10.1007/BF01060624
author: V. N. Temlyakov
published in: Ukrainian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Denote by \(T(Q_ n)\) the set of trigonometric polynomials \(t(x)=\sum_{| k| \in Q_ n}c_ ke^{i(k,x)},\) where \(x=(x_ 1,...,x_ d)\in {\mathbb{R}}^ d,\quad k=(k_ 1,...,k_ d)\) are vectors with integral coordinates, \(| k| =(| k_ 1|,...,| k_ d|);\) \[ Q_ n=\cup_{s\in \theta_ n}\{k:[2^{s_ j-1}]\leq k_ j\leq 2^{s_ j},\quad j=1,...,d\}, \] \[ \theta_ n=\{s=(s_ 1,...,s_ d):s_ j\quad are\quad nonnegative\quad integers,\quad | s_ 1| +...+| s_ d| \leq n\}. \] Polynomials of this form are called polynomials with harmonics of hyperbolic crosses. The author obtains an analogue of Jackson's theorem for approximation of continuous functions by trigonometric polynomials from hyperbolic crosses.
Mathematics Subject Classification ID: 42A10
Mathematics Subject Classification ID: 42B99
zbMATH DE Number: 4175665
zbMATH Keywords: harmonics of hyperbolic crosses
zbMATH Keywords: analogue of Jackson's theorem
zbMATH Keywords: approximation of continuous functions by trigonometric polynomials from hyperbolic crosses.
MaRDI profile type: Publication
cites work: Q3284729
cites work: Q3729529
cites work: Reconstruction of periodic functions of several variables with respect to the values in the nodes of number-theoretic nets
description: scientific article; zbMATH DE number 4175665