Wavelets with patchwise cancellation properties

We construct wavelets on general n-dimensional domains or manifolds via a domain decomposition technique, resulting in so-called composite wavelets. With this construction, wavelets with supports that extend to more than one patch are only continuous over the patch interfaces. Normally, this limited smoothness restricts the possibility for matrix compression, and with that the application of these wavelets in (adaptive) methods for solving operator equations. By modifying the scaling functions on the interval, and with that on the n-cube that serves as parameter domain, we obtain composite wavelets that have patchwise cancellation properties of any required order, meaning that the restriction of any wavelet to each patch is again a wavelet. This is also true when the wavelets are required to satisfy zeroth order homogeneous Dirichlet boundary conditions on (part of) the boundary. As a result, compression estimates now depend only on the patchwise smoothness of the wavelets that one may choose. Also taking stability into account, our composite wavelets have all the properties for the application to the (adaptive) solution of well-posed operator equations of orders 2t for t ∈ (− 1 2 , 3 2 ). 1. Motivation and background For some n′ ≥ n ≥ 1, let Ω be an n-dimensional manifold in Rn . We are interested in approximating the solution of an equation Lu = f , where for some Hilbert space H of functions on Ω, typically being a Sobolev space, with dual H ′, L : H → H ′ is boundedly invertible, and f ∈ H ′. When Ω is a domain in R, we think of the equation as being the result of a variational formulation of a boundary value problem, and when it is a true manifold, we have in mind an integral equation formulated on the boundary of an (n+ 1)-dimensional domain. Now let us assume that we have available a Riesz basis Ψ for H of wavelet type, where each wavelet is assumed to have the cancellation property of a certain order, meaning that, possibly after making some smooth transformation of coordinates, it is orthogonal to all polynomials of that order. Thinking of strongly elliptic problems, for any V ⊂ H spanned by some finite subset of the wavelets, we can approximate u by the Galerkin solution from V . This approach has two attractive features. First, since Ψ is a Riesz basis, the stiffness matrix with respect to the wavelet basis is well conditioned uniformly in V , allowing an efficient iterative solution. Second, for L being a singular integral Received by the editor October 8, 2004 and, in revised form, March 14, 2005. 2000 Mathematics Subject Classification. Prmary 46B15, 46E35, 65N55, 65T60.