Stable decomposition of homogeneous Mixed-norm Triebel–Lizorkin spaces

We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel–Lizorkin spaces in an anisotropic setting on R^d. The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel–Lizorkin spaces. In the second part of the paper we study nonlinear m-term approximation with the constructed basis in the mixed-norm setting, where the behavior, in general, for d≥2, is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for m-term approximation can still be derived.