Non-uniform painless decompositions for anisotropic Besov and Triebel-Lizorkin spaces - Cabrelli, C. (2013)
In this article we construct affine systems that provide a simultaneous atomic decomposition for a wide class of functional spaces including the Lebesgue spaces Lp(Rd), 1 < p < + ∞. 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 Λ-satisfying a certain spreadness condition but otherwise irregular-there exists a smooth window whose translations along the elements of Λ 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. © 2012 Elsevier Ltd.