Nonstationary Subdivision Schemes and Multiresolution Analysis

Nonstationary subdivision schemes consist of recursive refinements of an initial sparse sequence with the use of masks that may vary from one scale to the next finer one. This paper is concerned with both the convergence of nonstationary subdivision schemes and the properties of their limit functions. We first establish a general result on the convergence of such schemes to $C^\infty $ compactly supported functions. We show that these limit functions allow us to define a multiresolution analysis that has the property of spectral approximation. Finally, we use these general results to construct $C^\infty $ compactly supported cardinal interpolants and also $C^\infty $ compactly supported orthonormal wavelet bases that constitute Riesz bases for Sobolev spaces of any order.