Composite Wavelet Bases with Extended Stability and Cancellation Properties

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, which bases have been used, via a domain decomposition approach, 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, and, moreover, those wavelets with supports that extend to more than one patches generally have no cancellation properties. In this paper, we construct local, composite wavelets that are sufficiently close to being biorthogonal with respect to the standard L 2-scalar product, so that they generate Riesz bases for the Sobolev spaces H s for full range of s that is 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. 1. Introduction. The use of wavelet bases for solving operator equations, as partial differential equations or (boundary) integral equations, has a number of advantages , cf. [9, 3]. Assuming that the operator is symmetric, and, for H being some Hilbert space, H-bounded and H-coercive, and that the infinite collection of, properly scaled, wavelets generates a Riesz basis for H, the stiffness matrix in wavelet coordinates resulting from a Ritz-Galerkin discretization is well-conditioned uniformly in its size, guaranteeing a uniform rate of convergence of an iterative method. In case of a differential operator, this stiffness matrix is not truly sparse, but has the well-known " finger structure ". For multiplying with this matrix, however, one may switch to single-scale basis, with respect to which the stiffness matrix is sparse. For integral operators, the stiffness matrix with respect to both single-scale and wavelet basis is densely populated. Here the second important property of wavelets can be exploited of having vanishing moments or, …