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

Abstract

We provide a new closed form expression for the Geronimus polynomials on the unit circle and use it to obtain new results and formulas. Among our results is a universality result at an endpoint of an arc for polynomials orthogonal with respect to a Geronimus type weight on an arc of the unit circle. The key tool is a formula of McLaughlin for the n th power of a 2 × 2 matrix, which we use to derive convenient formulas for Geronimus polynomials.

Applications of a new formula for OPUC with periodic Verblunsky coefficients

“…If |∆(e iθ )| = 2 and e iθ is not a resonance, then Lemma 3.2 shows that |ϕ j (e i(θ+cσ j ) )| = O(j) as j → ∞ uniformly in c in compact subsets of C, while K kp−1 (e iθ , e iθ ; µ) grows like a constant times k 3 as k → ∞ (see also [5,Theorem 1.1]). Taken together, this implies (24) in this case. Lemma 3.8 and the calculations preceding it give us a complete proof of Theorem 3.1 in the case µ * = µ.…”

“…Since µ({e iθ }) = 0, we know that K kp−1 (e iθ , e iθ ; µ) → ∞ as k → ∞. If ∆(e iθ ) ∈ (−2, 2), then Lemmas 3.2 and 3.6 shows that {|ϕ j (e i(θ+cσ j ) )|} j∈N is bounded uniformly in j ∈ N and uniformly in c in compact subsets of C. This implies (24). The same reasoning applies if |∆(e iθ )| = 2 and e iθ is a resonance.…”

“…The analog of Theorem 2.1 on the real line was proven by de Jesus and Petronilho (see [8,Theorem 5.1] and also [6]). The special case p = 1 was considered in [24].…”

“…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.…”

“…These formulas for V and W will be important when stating our results in Section 3. As in [24], our results will be made possible by a new formula for the orthonormal polynomials {ϕ n } ∞ n=0 in terms of the Chebyshev polynomials of the second kind. These polynomials have many closed form expressions, but the one that will be most useful for us is…”