Irregular Wavelets and Anisotropic Smoothness spaces: a global characterization

CABRELLI, CARLOS A.; MOLTER, URSULA M.; ROMERO, JOSE LUIS

Cordoba

Congreso; IV Congreso Latinoamericano de Matematica; 2012

UMALCA

In this talk we show the construction of affine systems that provide atomic decompositions for a wide class of functional spaces including the Lebesgue spaces Lp(R), 1 < p < +∞. The novelty and the difficulty of this construction is that it allows for no-lattice transla- tions. In particular we show a way to explicitly obtain a dual system that provides the reconstruction formulas. We have two different decompositions. In one of then we use band-limited generators, and in the other one the window that we obtain is smooth and compactly supported. We prove that for every arbitrary expansive matrix A and any set Λ - satisfying certain spreadness condition but otherwise irregular- there exists a smooth compactly supported window whose translations along the elements of Λ and dilations by powers of A provide an atomic decomposition for a whole range of the anisotropic Triebel-Lizorkin spaces. To derive this result we start with a known general ?painless? construction, which gives a band-limited generator that provides an affine system for arbitrary matrix A and set Λ. We then prove the existence of a band-limited dual system of molecules who provide an atomic decomposition for a whole class of anisotropic Besov-Triebel-Lizorkin spaces.