Multiresolution weighted norm equivalences and applications

Abstract

We establish multiresolution norm equivalences in weighted spaces $L^2_w$ ((0,1)) with possibly singular weight functions $w(x) \geq 0$ in (0,1). Our analysis exploits the locality of the biorthogonal wavelet basis and its dual basis functions. The discrete norms are sums of wavelet coefficients which are weighted with respect to the collocated weight function $w(x)$ within each scale. Since norm equivalences for Sobolev norms are by now well-known, our result can also be applied to weighted Sobolev norms. We apply our theory to the problem of preconditioning $p$-Version FEM and wavelet discretizations of degenerate elliptic and parabolic problems from finance.