Sobolev exponents of Butterworth refinable functions

Computing the Smoothness Exponent of a Symmetric Multivariate Refinable Function

Smoothness and symmetry are two important properties of a refinable function. It is known that the Sobolev smoothness exponent of a refinable function can be estimated by computing the spectral radius of a certain finite matrix which is generated from a mask. However, the increase of dimension and the support of a mask tremendously increases the size of the matrix and therefore makes the comput...

Characterization of Smoothness of Multivariate Refinable Functions in Sobolev Spaces

Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the smoothness properties of multivariate refinable functions in Sobolev spaces. We characterize the optimal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant s...

Vector cascade algorithms and refinable function vectors in Sobolev spaces

In this paper we shall study vector cascade algorithms and refinable function vectors with a general isotropic dilation matrix in Sobolev spaces. By investigating several properties of the initial function vectors in a vector cascade algorithm, we are able to take a relatively unified approach to study several questions such as convergence, rate of convergence and error estimate for a perturbed...

Computing the Sobolev Regularity of Refinable Functions by the Arnoldi Method

Computing the Sobolev Regularity of Refinable Functions by the Arnoldi

The recent paper [J. Approx. Theory, 106 (2000), pp. 185–225] provides a complete characterization of the L2-smoothness of a refinable function in terms of the spectrum of an associated operator. Based on this theory, we devise in this paper a numerically stable algorithm for calculating that smoothness parameter, employing the deflated Arnoldi method to this end. The algorithm is coded in Matl...