Invariant polytopes of linear operators with applications to regularity of wavelets and of subdivisions

TL;DR

This paper extends an invariant polytope algorithm to a broader class of matrix sets, enabling the computation of joint spectral radii and regularity properties of wavelets and subdivision schemes, solving key open problems.

Contribution

It generalizes the invariant polytope algorithm to matrices with finitely many spectrum maximizing products, with proven convergence criteria and practical applications.

Findings

Determined the regularity of the Butterfly subdivision scheme for various parameters.

Computed the H"older exponent of high-order Daubechies wavelets.

Proved the algorithm's convergence for a wider class of matrix sets.

Abstract

We generalize the recent invariant polytope algorithm for computing the joint spectral radius and extend it to a wider class of matrix sets. This, in particular, makes the algorithm applicable to sets of matrices that have finitely many spectrum maximizing products. A criterion of convergence of the algorithm is proved. As an application we solve two challenging computational open problems. First we find the regularity of the Butterfly subdivision scheme for various parameters $\omega$. In the "most regular" case $\omega = \frac{1}{16}$, we prove that the limit function has H\"older exponent $2$ and its derivative is "almost Lipschitz" with logarithmic factor $2$. Second we compute the H\"older exponent of Daubechies wavelets of high order.