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arXiv · 1108.2748

Non-uniform painless decompositions for anisotropic Besov and Triebel-Lizorkin spaces

Carlos Cabrelli·Ursula Molter·José Luis Romero
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Abstract

In this article we construct affine systems that provide a simultaneous atomic decomposition for a wide class of functional spaces including the Lebesgue spaces $L^p(\Rdst)$, $1<p<+\infty$. The novelty and difficulty of this construction is that we allow for non-lattice translations. We prove that for an arbitrary expansive matrix $A$ and any set $\Lambda$ - satisfying a certain spreadness condition but otherwise irregular- there exists a smooth window whose translations along the elements of $\Lambda$ and dilations by powers of $A$ provide an atomic decomposition for the whole range of the anisotropic Triebel-Lizorkin spaces. The generating window can be either chosen to be bandlimited or to have compact support. To derive these results we start with a known general "painless" construction that has recently appeared in the literature. We show that this construction extends to Besov and Triebel-Lizorkin spaces by providing adequate dual systems.

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Carlos Cabrelli, Ursula Molter, José Luis Romero. 2011-08-13. Non-uniform painless decompositions for anisotropic Besov and Triebel-Lizorkin spaces. https://arxiv.org/abs/1108.2748

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