Composite Wavelet Bases with Extended Stability and Cancellation Properties

Abstract

The efficient solution of operator equations using wavelets requires that they generate a Riesz basis for the underlying Sobolev space and that they have cancellation properties of a sufficiently high order. Suitable biorthogonal wavelets were constructed on reference domains as the n‐cube. Via a domain decomposition approach, these bases have been used as building blocks to construct biorthogonal wavelets on general domains or manifolds, where, in order to end up with local wavelets, biorthogonality was realized with respect to a modified $L_2$‐scalar product. The use of this modified scalar product restricts the application of these so‐called composite wavelets to problems of orders strictly larger than $-1$. Moreover, those wavelets with supports that extend to more than one patch generally have no cancellation properties. In this paper, we construct local, composite wavelets that are close to being biorthogonal with respect to the standard $L_2$‐scalar product. As a consequence, they generate Riesz bases for the Sobolev spaces $H^s$ for the full range of s allowed by the continuous gluing of functions over the patch interfaces, the properties of the primal and dual approximation spaces on the reference domain, and, in the manifold case, by the regularity of the manifold. Moreover, all these wavelets have cancellation properties of the full order induced by the approximation properties of the dual spaces on the reference domain. We illustrate our findings by a concrete realization of wavelets on a perturbed sphere.