Almost diagonal matrices and Besov-type spaces based on wavelet expansions

Abstract: This paper is concerned with problems in the context of the theoretical foundation of adaptive (wavelet) algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$ on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of $d$-dimensional manifolds and investigate some analytical properties (such as, e.g., embeddings and best $n$-term approximation rates) of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems $\Psi$ on domains or manifolds $\Gamma$ which admit a decomposition into smooth patches actually generate the same Besov-type function spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$, provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity (compared to the smoothness parameter $\alpha$ of the space). For this purpose, a theory of almost diagonal matrices on related sequence spaces $b^\alpha_{p,q}(\nabla)$ of Besov type is developed. Keywords: Besov spaces, wavelets, localization, sequence spaces, adaptive methods, non-linear approximation, manifolds, domain decomposition.