Two universality results for polynomial reproducing kernels

Abstract

Abstract. We prove two new universality results for polynomial reproducing kernels of compactly supported measures. The first applies to measures on the unit circle with a jump and a singularity in the weight at 1 and the second applies to area-type measures on a certain disconnected polynomial lemniscate. In both cases, we apply methods developed by Lubinsky to obtain our results.

Universality at an endpoint for orthogonal polynomials with Geronimus-type weights

“…The proof of Lemma 4.5 is very much analogous to the proof of [23,Lemma 2.8], so we will not present the details here. It is based on Christoffel functions and relies heavily on ideas from the proof of [18,Theorem 7]. )…”

Applications of a new formula for OPUC with periodic Verblunsky coefficients

“…The key remaining part of the proof of Theorem 3.1 is to show that lim n→∞ K n (e i(θ−aσn) , e i(θ−aσn) ; µ) K n (e i(θ−aσn) , e i(θ−aσn) ; µ * ) = w(θ) and the convergence is uniform for a in compact subsets of C. Proving this is where we use the fact that the measure µ * is regular and involves interpreting the diagonal reproducing kernel in terms of Christoffel functions and a localization argument. The details appear in many places, such as [10,14,23], so we omit the lengthy calculations. Due to the possible presence of mass points in the gaps of supp(µ ac ), we mention that the Erdős-Turán criterion (see [31, page 101]) and [27,Theorem 11.3.2] imply that not only is µ * regular, but so is its restriction to supp(µ ac ).…”

“…With these tools in hand, one will have no trouble adapting the method used in [10,12,23,24] to complete the proof of Theorem 3.1.…”

“…A desire to study limits of the above form comes from random matrix theory, where such limits can be useful in calculating eigenvalue correlation functions. There are many examples of measures on the unit circle for which these limits can be calculated (see [2,10,12,23,24]), but the case of a measure supported on several arcs of the unit circle has not been previously considered. That is the case we aim to consider here and we note that the analogous results on the real line have been proven by Totik in [32] and by Danka in [4].…”

“…Lemma 3.8 and the calculations preceding it give us a complete proof of Theorem 3.1 in the case µ * = µ. The general case now follows standard perturbative methods as in [10,12,23,24]. Due to the well-known method of proof, we provide only a sketch of the ideas.…”