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

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 ( R d ) , 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.