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

Nicola Guglielmi, Vladimir Yu. Protasov

Published 2015-02-04Version 1

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.