Criteria for Hierarchical Bases in Sobolev Spaces

Several approaches to solving elliptic problems numerically are based on hierarchical Riesz bases in Sobolev spaces. WSe are interested in determining the exact range of Sobolev exponents for which a system of compactly supported functions does form such a Reisz basis. When the system derives from a multiresolution analysis, this involves determing the smoothness of the dual system. The elements of the dual system typically consist of non-compactly supported functions, whose smoothness can be treated by extending the results of [9, 26, 10]. We show how to determine the exact range of Sobolev exponents in the multivariate case, both theoretically and numerically. This involves determining the spectral properties of transfer operators. This technique is applied to several bases deriving from linear finite elements which have been proposed is the literature. For Stevenson's hierarchical basis, we find that it forms a Riesz basis in H sup s (R sup d) for -0.990236...