Hyperbolic Wavelet Analysis of Classical Isotropic and Anisotropic Besov–Sobolev Spaces

Abstract In this paper we introduce new function spaces which call anisotropic hyperbolic Besov and Triebel-Lizorkin spaces. Their definition is based on a Littlewood-Paley analysis involving an anisotropy vector only occurring in the smoothness weights. Such provide general natural setting order to understand what kind of can be described using wavelets (in literature also sometimes called tensor-product wavelets), wavelet class hitherto has been mainly used characterize dominating mixed smoothness. A centerpiece our present work are characterizations these transform. Hereby treat both, standard approach systems equipped with sufficient smoothness, decay, vanishing moments, but very simple basic Haar system. The second major question pursue relationship between novel classical Besov–Lizorkin-Triebel scales. As results show, general, both approaches resolve do not coincide. However, Sobolev range case, providing link apply newly obtained setting. particular, allows for detecting anisotropies via coefficients universal basis, without need adaption basis or a-priori knowledge anisotropy.