Criteria for Hierarchical Bases in Sobolev Spaces

Abstract

AbstractSeveral approaches to solving elliptic problems numerically are based on hierarchical Riesz bases in Sobolev spaces. We are interested in determining the exact range of Sobolev exponents for which a system of compactly supported functions derived from a multiresolution analysis forms such a Riesz basis. This involves determining the smoothness of the dual system. The elements of the dual system typically consist of noncompactly supported functions, whose smoothness can be treated by extending the results of 7, 9, and 22. We show how to determine the exact range of Sobolev exponents in the multivariate case, both theoretically and numerically, from spectral properties of transfer operators. This technique is applied to several bases deriving from linear finite elements which have been proposed in the literature. For 29hierarchical basis, we find that it forms a Riesz basis in Hs(Rd) for −0.990236…<s<3/2