Computing the Sobolev regularity of refinable functions by the Arnoldi method

The recent paper [J. Approx. Theory, 106 (2000), pp. 185-225] provides a co mplete characterization of the L-2-smoothness of a refinable function in te rms of the spectrum of an associated operator. Based on this theory, we dev ise in this paper a numerically stable algorithm for calculating that smoot hness parameter, employing the deflated Arnoldi method to this end. The alg orithm is coded in Matlab, and details of the numerical implementation are discussed, together with some of the numerical experiments. The algorithm i s designed to handle large masks, as well as masks of refinable functions w ith unstable shifts. This latter case is particularly important, in view of the recent developments in the area of wavelet frames.