Lommel polynomials and explicitly solvable prediction problems on the unit circle

Abstract

To each finite symmetric measure $\sigma$ on the real line, with support a compact subset of $(-2,2)$, we associate a measure $\mu$ on the unit circle by transporting mass at $\pm x$ to $e^{\pm i\theta(x)}$, $\theta(x)=2\arcsin(x/2)$. The linear prediction errors, Verblunsky coefficients, and Toeplitz determinants of $\mu$ are then expressed through the orthogonal polynomial data of $\sigma$ at the single edge point $x=2$. In particular $E_m(\mu)=\tfrac12 t_m\|P_m\|_\sigma^2$ with $t_m=P_{m+1}(2)/P_m(2)$. Under a condition on the first coefficients, decay of the recurrence coefficients of $\sigma$ forces the $t_m$ to increase from $t_1$ onward, a Tur\'an-type monotonicity in the degree placing every prediction margin of $\mu$ past the first above the corresponding coefficient of $\sigma$. Taking $\sigma$ to be the Lommel-polynomial measures, with atoms at rescaled reciprocals of the zeros of the Bessel function $J_\nu$, yields a two-parameter family of purely atomic circle measures with a normalized determinant limit equal to a value of $J_\nu$.