Decomposition systems for function spaces

G. Kyriazis¹

¹ Department of Mathematics and Statistics University of Cyprus P.O. Box 20537 1678 Nicosia, Cyprus

Studia Mathematica, Tome 157 (2003) no. 2, pp. 133-169

Résumé

Let ${\mit\Theta}:=\{\theta_I^e: e\in E,\, I\in D\}$ be a decomposition system for $L_2({\mathbb R}^d)$ indexed over $D$, the set of dyadic cubes in ${\mathbb R}^d$, and a finite set $E$, and let $\widetilde{\mit\Theta}:=\{\widetilde\theta^e_I: e\in E,\, I\in D\}$ be the corresponding dual functionals. That is, for every $f\in L_2({\mathbb R}^d)$, $f=\sum_{e\in E}\sum_{I\in D}\def\inpro#1#2{\langle#1,#2\rangle} \inpro{f}{\widetilde\theta^e_I}\theta_I^e$. We study sufficient conditions on ${\mit\Theta},\widetilde{\mit\Theta}$ so that they constitute a decomposition system for Triebel–Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution $f$ in these spaces by the size of the coefficients $\def\inpro#1#2{\langle#1,#2\rangle}\inpro{f}{\widetilde\theta^e_I}$, $e\in E$, $I\in D$. Typical examples of such decomposition systems are various wavelet-type unconditional bases for $L_2({\mathbb R}^d)$, and more general systems such as affine frames.

Détail

DOI : 10.4064/sm157-2-3

Keywords: mit theta theta decomposition system mathbb indexed set dyadic cubes mathbb finite set widetilde mit theta widetilde theta corresponding dual functionals every mathbb sum sum def inpro langle rangle inpro widetilde theta theta study sufficient conditions mit theta widetilde mit theta constitute decomposition system triebel lizorkin besov spaces moreover these conditions allow characterize membership distribution these spaces size coefficients def inpro langle rangle inpro widetilde theta typical examples decomposition systems various wavelet type unconditional bases mathbb general systems affine frames