Piecewise Tensor Product Wavelet Bases by Extensions and Approximation Rates

Abstract

Following [Studia Math., 76(2) (1983), pp. 1-58 and 95-136] by Z. Ciesielski and T. Figiel and [SIAM J. Math. Anal., 31 (1999), pp. 184-230] by W. Dahmen and R. Schneider, by the application of extension operators we construct a basis for a range of Sobolev spaces on a domain $\Omega$ from corresponding bases on subdomains that form a non-overlapping decomposition. As subdomains, we take hypercubes, or smooth parametric images of those, and equip them with tensor product wavelet bases. We prove approximation rates from the resulting piecewise tensor product basis that are independent of the spatial dimension of $\Omega$. For two- and three-dimensional polytopes we show that the solution of Poisson type problems satisfies the required regularity condition. The dimension independent rates will be realized numerically in linear complexity by the application of the adaptive wavelet-Galerkin scheme