Product Besov and Triebel-Lizorkin Spaces with Application to Nonlinear Approximation

The Littlewood-Paley theory of homogeneous product Besov and Triebel-Lizorkin spaces is developed in the spirit of the phi-transform of Frazier and Jawerth. This includes the frame characterization of the product Besov and Triebel-Lizorkin spaces and the development of almost diagonal operators on these spaces. The almost diagonal operators are used to obtain product wavelet decomposition of the product Besov and Triebel-Lizorkin spaces. The main application of this theory is to nonlinear m-term approximation from product wavelets in L-p and Hardy spaces. Sharp Jackson and Bernstein estimates are obtained in terms of product Besov spaces.