The growth of polynomials orthogonal on the unit circle with respect to a weight $w$ that satisfies $w,w^{-1}\in L^\infty( {\mathbb{T}})$

Abstract

We consider the polynomials tφnpz, wqu orthogonal on the circle with respect to a weight w that satisfies w, w´1 P L 8 pTq and show that }φnpe iθ , wq} L 8 pTq can grow in n at a certain rate.

Mate-Nevai-Totik theorem for Krein systems

“…Construct a Krein system with the coefficient a(r) = − 1 2 p(r/2)+ i 2 q(r/2), where p, q are the potentials in equation (1). Uniqueness theorem and (12) give that functions ϕ and ψ defined by (10) and (11) respectively are the unique solutions of equation (1). Since potentials p, q are in L 2 (R + ), coefficient a is also in L 2 (R + ).…”

Continuity of weighted operators, Muckenhoupt $A_p$ weights, and Steklov problem for orthogonal polynomials

“…Thus, log w P L 1 pTq as well. This means that µ : dµ " w 2π dθ belongs to Szegő class of measures and, consequently (see [9]),…”

“…From lemma 3.2 and lemma 3.4, we get that p cr ptq ą 2 and lim tÑ1 p cr ptq " 8. To show that p cr ptq Ñ 2 when t Ñ 8, we use the following result established in [9], theorem 3.2: given any t ą 2, there is a weight w that satisfies 1 w t and a subsequence tk n u such that…”

Randomized Verblunsky Parameters in Steklov's Problem

“…This question was answered in the affirmative in [4,5], which imposed w, w −1 ∈ L ∞ (T) and w, w −1 ∈ BMO(T), respectively, although both consider only polynomials orthogonal with respect to absolutely continuous measures. Both conditions are sufficient to break the n 1/2 barrier, and lower bounds were established in [4] showing that its results are sharp in some regimes.…”