On an orthogonal bivariate trigonometric Schauder basis for the space of continuous functions
type: Item
instance of: scholarly article
title: On an orthogonal bivariate trigonometric Schauder basis for the space of continuous functions
zbMATH Open document ID: 1409.42025
author: Nadiia Derevianko
author: V. V. Myronyuk
author: Jürgen Prestin
published in: Journal of Approximation Theory
publication date: 2019-01-18T00:00:00Z
review text: The main result of the article is the construction of an orthogonal trigonometric Schauder basis in the space of all continuous functions on the torus \(C(T^2)\) with the smallest growth of the polynomial degree. This is achieved by using ideas involving a dyadic anisotropic periodic multiresolution analysis. Let \(T^2=[-\pi, \pi)^2\) be the two-dimensional torus and for any vector \(\mathbf{x}=(x_1, x_2)\in \mathbb R^2,\) set \[ \left\| \mathbf{x}\right\|_{\ell^q} =\begin{cases} \left( |x_1|^q +|x_2|^q \right)^{1/q}& 1\leq q< \infty \\ \max(|x_1|, |x_2| ) & q=\infty .\end{cases} \] The \(q\)-degree of a trigonometric polynomial \(t\) of two variables is defined as \[ \text{deg}_q t=\min \left\{ n\in \mathbb{N} : t(\mathbf{x}) =\sum_{\left\| \mathbf{k} \right\|_{\ell_q}\leq n }c_{\mathbf{k} } e^{ i(k_1 x_1 +k_2 x_2)}, c_{\mathbf{k} }\in \mathbb{C}, \mathbf{k} \in \mathbb Z^2 \right\} . \] For any \(f\in C(T^2),\) the operator \(S_\mu f\) is defined as \[ S_\mu f=\sum_{k=1}^\mu \langle f, t_k \rangle t_k , \] where \(t_k\) is a trigonometric polynomial, \(\mu, k\in\mathbb{N}.\) For any set \(A\subset \mathbb Z^2,\) let \[ T_A =\left\{t: t(\mathbf{x})=\sum_{\mathbf{k}\in A} c_{\mathbf{k} } e^{ i(k_1 x_1 +k_2 x_2)}, c_{\mathbf{k} }\in \mathbb{C} \right\} . \] The distance between \(f\) and \(T_A\) is defined as \[ E_A(f)_\infty =\inf_{t\in T_A} \left\| f-t\right\|_\infty. \] The authors construct a polynomial system \(\left\{ t_k: k\in \mathbb{N} \right\}\) that is an orthonormal Schauder basis in \(C(T^2)\) such that for any \(f\in C(T^2),\) the following relation holds \[ \left\| f-S_\mu f\right\|_\infty\leq \left( 1+C(\epsilon) \right) E_A(f)_\infty, \] where \(C(\epsilon)>0.\) It is also shown that the orthogonal projection operator \(S_\mu\) for the functions \(\left\{ t_k: k\in \mathbb{N} \right\}\) acting as an operator from \(C(T^2)\) into itself is uniformly bounded for all \(\mu \in \mathbb{N},\) i.e., \[ \left\|S_\mu \right\|\leq C(\epsilon) , \quad 0< C(\ep…
reviewed by: Ahmed I. Zayed
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 65T40
zbMATH DE Number: 7003505
zbMATH Keywords: bivariate trigonometric Schauder basis
zbMATH Keywords: anisotropic periodic multiresolution analysis
MaRDI profile type: Publication
full work available at URL: j.jat.2017.05.007
OpenAlex ID: W2734350273
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DOI: 10.1016/J.JAT.2017.05.007
description: scientific article; zbMATH DE number 7003505