Computing the Sobolev Regularity of Refinable Functions by the Arnoldi Method

The recent paper [Approx. Theory,(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 Matlab, and details of the numerical implementation are discussed, together with some of the numerical experiments. The algorithm is designed to handle large masks, as well as masks of refinable functions with unstable shifts. This latter case is particularly important, in view of the recent developments in the area of wavelet frames.

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