On the Construction of Bases and Frames with Applications

Kamen G. Ivanov (), Gerard Kerkyacharian (), George Kyriazis () and Pencho Petrushev ()
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Kamen G. Ivanov: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
Gerard Kerkyacharian: University Paris
George Kyriazis: University of Cyprus, Department of Mathematics and Statistics
Pencho Petrushev: University of South Carolina, Department of Mathematics

A chapter in Multiscale, Nonlinear and Adaptive Approximation II, 2024, pp 347-381 from Springer

Abstract: Abstract This article is a survey of results on the construction of bases and frames for Besov and Triebel-Lizorkin spaces (B- and F-spaces) in various classical and nonclassical settings.We first review our small-perturbation method for construction of bases for B- and F-spaces on $$ \mathbb{R}^{\textit{d}} $$ based on wavelets and show some of its applications to nonlinear approximation. Then we describe the construction of frames for B- and F-spaces in the classical setting on $$ \mathbb{R}^{\textit{d}} $$ , on the sphere in $$ \mathbb{R}^{\textit{d}} $$ , and in a general setting that covers a number of classical and nonclassical setups such as on $$ \mathbb{R}^{\textit{d}} $$ , the interval, sphere, ball, simplex as well as Riemannian manifolds and Lee groups. Finally, we present a small perturbation method for construction of frames for Band F-spaces in a general setting. Several applications of this scheme are presented.

Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-75802-7_16

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DOI: 10.1007/978-3-031-75802-7_16

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