Computing the Sobolev regularity of refinable functions by the Arnoldi method (Q2719202)

Let a function \( \varphi \in L^2 ({\mathbb R}^d) \) be given by its refinement equation NEWLINE\[NEWLINE \varphi (x) = |\det s |\sum_{j \in {\mathbb Z}^d} a(j) \varphi (sx-j), \qquad x \in {\mathbb R}^d , NEWLINE\]NEWLINE where \(s\) is a \(d \times d\) integer matrix with \( s^{\ast} s = \lambda I\) for some \( \lambda > 1\), and \(a = (a_j)_{j \in {\mathbb Z}^d} \) is a finite sequence. NEWLINENEWLINENEWLINEThe Sobolev-regularity of refinable functions can be characterized in terms of a linear operator (the transfer operator defined by \(a\)) restricted to a certain space \( H \) of trigonometric polynomials [cf. \textit{A. Ron} and \textit{Z. Shen}, J. Approximation Theory 106, No. 2, 185-225 (2000; Zbl 0966.42026)]. NEWLINENEWLINENEWLINEIn order to find a numerically stable algorithm to compute the smoothness parameter of \( \varphi \), the space \( H \) needs to be characterized such that on the one hand it applies to a wide range of refinement equations, and on the other hand it is computational verifiable. NEWLINENEWLINENEWLINEThe presented algorithm solves this problem by computing eigenvalues and eigenvectors of the transfer operator in \( H \). The occurring large eigenproblems are solved with a variation of the Arnoldi method. NEWLINENEWLINENEWLINEThe algorithm is coded in Matlab and numerical experiments are presented. The algorithm can handle large refinement masks and masks of refinable functions with unstable shifts.