Positive definite periodic functions and polynomials orthogonal on the unit circle with dense point spectrum

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.