WAVELETS IN SOBOLEV SPACES
Abstract
In the first part, we present two constructions of biorthogonal bases of wavelets in Sobolev spaces Hm() of integer order. On one hand, we give a Sobolev version of the L2 biorthogonal wavelets bases construction of A. Cohen, I. Daubechies, J. C. Feauveau, and on the other hand, a Sobolev version of the L2 classical construction of C. K. Chui and J. Z. Wang.
In the second part, we show how hierarchical periodic spline spaces can be used to approximate solutions for a large class of pseudodifferential equations on boundaries of smooth open subsets of 2. We give a simple proof of the characterization of the coercivity condition, which leads to relations on the meshes and order of the splines easy to handle. Then we endow the periodic Sobolev space Hs1-per() with a norm equivalent to the natural one which makes the Galerkin system of Céa lemma equivalent to a collocation system for high resolution levels. Test and trial spline bases are explicitly given. We investigate the asymptotic stability of such systems and we present some numerical experiments
1AMS Classification: 42C40, 46335, 41A15, 65L60