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1. 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 subarcs
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2. ; (Ed.)
Abstract. Let fj g1 j=1 be a sequence of distinct positive numbers. Let w be a nonnegative function, integrable on the real line. One can form orthogonal Dirichlet polynomials fng from linear combinations of n
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3. 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 ( 1-z^{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 .
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4. Endpoint masspoints
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5. 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→∞.
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6. (Ed.)
We determine the asymptotics for the variance of the num-ber of zeros of random linear combinations of orthogonal polynomials ofdegreenin subintervals[a;b]of the support of the underlying orthog-onality measure. We show that, asn!1, this variance is asymptotictocn, for some explicit constantc
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7. (Ed.)
Letbe 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, while0is positive and continuous. Letf'ngbe the orthonormal polynomials for. We show that for appropriaten2J,'n(n(1+zn))'n(n)n1is a normal family in compact subsets ofC. Usinguniversality limits, we show that limits of subsequences have the formez+C(ez
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