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Let be a measure on the unit circle that is regu lar in the sense of Stahl Totik, and Ullmann. Let f'ng be the orthonormal polynomials for and fzjng their zeros. Let be absolutely continuous in an arc of the unit circle, with 0 pos itive and continuous there. We show that uniform boundedness of the orthonormal polynomials in subarcsmore » « less

Let { s j } j = 1 n \left \{ s_{j}\right \} _{j=1}^{n} be positive integers. We show that for any 1 ≤ L ≤ n , 1\leq L\leq n, ‖ ∏ j = 1 n ( 1 − z s j ) ‖ L ∞ (  z  = 1 ) ≥ exp ( 1 2 e L ( s 1 s 2 … s L ) 1 / L ) . \begin{equation*} \left \Vert \prod _{j=1}^{n}\left ( 1z^{s_{j}}\right ) \right \Vert _{L_{\infty }\left ( \left \vert z\right \vert =1\right ) }\geq \exp \left ( \frac {1}{2e}\frac {L}{\left ( s_{1}s_{2}\ldots s_{L}\right ) ^{1/L}}\right ) . \end{equation*} In particular, this gives geometric growth if a positive proportion of the { s j } \left \{ s_{j}\right \} are bounded. We also show that when the { s j } \left \{ s_{j}\right \} grow regularly and faster than j ( log j ) 2 + ε j\left ( \log j\right ) ^{2+\varepsilon } , some ε > 0 \varepsilon >0 , then the norms grow faster than exp ( ( log n ) 1 + δ ) \exp \left ( \left ( \log n\right ) ^{1+\delta }\right ) for some δ > 0 \delta >0 .more » « less

Let ν be a positive measure supported on [1,1], with infinitely many points in its support. Let {p_{n}(ν,x)}_{n≥0} be its sequence of orthonormal polynomials. Suppose we add masspoints at ±1, giving a new measure μ=ν+Mδ₁+Nδ₋₁. How much larger can p_{n}(μ,0) be than p_{n}(ν,0)? We study this question for symmetric measures, and give more precise results for ultraspherical weights. Under quite general conditions, such as ν lying in the Nevai class, it turns out that the growth is no more than 1+o(1) as n→∞.more » « less

null (Ed.)We determine the asymptotics for the variance of the number of zeros of random linear combinations of orthogonal polynomials ofdegreenin subintervals[a;b]of the support of the underlying orthogonality measure. We show that, asn!1, this variance is asymptotictocn, for some explicit constantcmore » « less

null (Ed.)Letbe a positive measure on the unit circle that is regularin the sense of Stahl, Totik, and Ullmann. Assume that in some subarcJ,is absolutely continuous, while0is positive and continuous. Letf'ngbe the orthonormal polynomials for. We show that for appropriaten2J,'n(n(1+zn))'n(n)n1is a normal family in compact subsets ofC. Usinguniversality limits, we show that limits of subsequences have the formez+C(ezmore » « less