Decomposition systems for function spaces
Volume 157 / 2003
Abstract
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.
