Christoffel formula for kernel polynomials on the unit circle

Abstract

Given a nontrivial positive measure μ on the unit circle T, the associated Christoffel–Darboux kernels are Kn(z,w;μ)=∑k=0 nφk(w;μ)¯φk(z;μ), n≥0, where φk(⋅;μ) are the orthonormal polynomials with respect to the measure μ. Let the positive measure ν on the unit circle be given by dν(z)=|G2m(z)|dμ(z), where G2m is a conjugate reciprocal polynomial of exact degree 2m. We establish a determinantal formula expressing {Kn(z,w;ν)}n≥0 directly in terms of {Kn(z,w;μ)}n≥0. Furthermore, we consider the special case of w=1; it is known that appropriately normalized polynomials Kn(z,1;μ) satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters {cn(μ)}n=1 ∞ and {gn(μ)}n=1 ∞, with 0<gn<1 for n≥1. The double sequence {(cn(μ),gn(μ))}n=1 ∞ characterizes the measure μ. A natural question about the relation between the parameters cn(μ), gn(μ), associated with μ, and the sequences cn(ν), gn(ν), corresponding to ν, is also addressed. Finally, examples are considered, such as the Geronimus weight (a measure supported on an arc of T), a measure for which the Christoffel–Darboux kernels, with w=1, are given by basic hypergeometric polynomials and a measure for which the orthogonal polynomials and the Christoffel–Darboux kernels, again with w=1, are given by hypergeometric polynomials.