Christoffel formula for kernel polynomials on the unit circle

Abstract

Given a nontrivial positive measure $\mu$ on the unit circle, the associated Christoffel-Darboux kernels are $K_n(z,w;\mu )={\sum }_{k=0}^n\overline{{\phi }_k(w;\mu )}{\phi }_k(z;\mu )$, $n\ge 0$, where ${\phi }_k(\cdot ;\mu )$ are the orthonormal polynomials with respect to the measure $\mu$. Let the positive measure $\nu$ on the unit circle be given by $d\nu (z)=|G_{2m}(z)|d\mu (z)$, where $G_{2m}$ is a conjugate reciprocal polynomial of exact degree $2m$. We establish a determinantal formula expressing $\{K_n(z,w;\nu ){\}}_{n\ge 0}$ directly in terms of $\{K_n(z,w;\mu ){\}}_{n\ge 0}$. Furthermore, we consider the special case of $w=1$; it is known that appropriately normalized polynomials $K_n(z,1;\mu )$ satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters $\{c_n(\mu ){\}}_{n=1}^{\infty }$ and $\{g_n(\mu ){\}}_{n=1}^{\infty }$, with $0<g_n<1$ for $n\ge 1$. The double sequence $\{(c_n(\mu ),g_n(\mu )){\}}_{n=1}^{\infty }$ characterizes the measure $\mu$. A natural question about the relation between the parameters $c_n(\mu )$, $g_n(\mu )$, associated with $\mu$, and the sequences $c_n(\nu )$, $g_n(\nu )$, corresponding to $\nu$, is also addressed. Finally, examples are considered, such as the Geronimus weight (a measure supported on an arc of the unit circle), a class of measures given by basic hypergeometric functions, and a class of measures with hypergeometric orthogonal polynomials.

References in corpus (5)

• Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas
• Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure
• CMV biorthogonal Laurent polynomials: Christoffel formulas for Christoffel and Geronimus perturbations
• Two families of orthogonal polynomials on the unit circle from basic hypergeometric functions
• CMV biorthogonal Laurent polynomials. II: Christoffel formulas for Geronimus-Uvarov perturbations