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 \(\varphi \)-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.