Seminars: A. S. Zhedanov, Positive definite periodic functions and polynomials orthogonal on the unit circle with dense point spectrum

SEMINARS


Complex Approximations, Orthogonal Polynomials and Applications (CAOPA) November 15, 2021 20:00, Moscow, online via Zoom at 17:00 GMT (=12:00 EST=18:00 CET=20:00 Msk)

Positive definite periodic functions and polynomials orthogonal on the unit circle with dense point spectrum A. S. Zhedanov^ab ^a Renmin University of China ^b Université de Montréal, Centre de Recherches Mathématiques
https://youtu.be/nHvONOeE3uA Abstract: We demonstrate that any positive definite periodic function $f(x)$ generates a set of polynomials orthogonal on the unit circle (OPUC) with dense point spectrum. Explicit examples of OPUC arise if $f(x)$ coincides with one of two Jacobi elliptic functions: $\textrm{cn}(x;k)$ or $\textrm{dn}(x,k)$. These OPUC have a simple explicit expression in terms of elliptic hypergeometric functions. A more elementary example corresponds to wrapped geometric distribution on the unit circle . In this case OPUC are expressed in terms of a basic hypergeometric function. In all the above cases corresponding OPUC satisfy remarkable “classical” properties. Language: English