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1. We show that universality limits and other bounds imply pointwise asymptotics for orthonormal polynomials at the endpoints of the interval of orthonormality. As a consequence, we show that if μ is a regular measure supported on [−1, 1], and in a neighborhood of 1, μ is absolutely continuous, while for some α > −1, μ (t) = h (t)(1 − t) α, where h (t) → 1 as t → 1−, then the corresponding orthonormal polynomials {pn} satisfy the asymptotic limn→∞pn1 − z22n2pn (1) = J∗α (z)J∗α (0) uniformly in compact subsets of the plane. Here J∗α (z) = Jα (z) /zα is the normalized Bessel function of order α. These are by far the most general conditions for such endpoint asymptotics
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2. We prove that for most entire functions f in the sense of category, a strong form of the Baker-Gammel-Wills Conjecture holds. More precisely, there is an inÖnite sequence S of positive integers n, such that given any r > 0, and multipoint PadÈ approximants Rn to f with interpolation points in fz : jzj  rg, fRngn2S converges locally uniformly to f in the plane. The sequence S does not depend on r, nor on the interpolation points. For entire functions with smooth rapidly decreasing coe¢ cients, full diagonal sequences of multipoint PadÈ approximants converge.
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3. We prove that for most entire functions f in the sense of category, a strong form of the Baker-Gammel-Wills Conjecture holds. More precisely, there is an infinite sequence S of positive integers n, such that given any r>0, and multipoint Padé approximants R_{n} to f with interpolation points in {z:|z|≤r}, {R_{n}}_{n∈S} converges locally uniformly to f in the plane. The sequence S does not depend on r, nor on the interpolation points. For entire functions with smooth rapidly decreasing coefficients, full diagonal sequences of multipoint Padé approximants converge.
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4. We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only $(2/\pi + o(1))\log{n}$ expected real zeros in terms of the degree $n$. If the basis is given by the orthonormal polynomials associated with a compactly supported Borel measure on the real line, or associated with a Freud weight, then random linear combinations have $n/\sqrt{3} + o(n)$ expected real zeros. We prove that the same asymptotic relation holds for all random orthogonal polynomials on the real line associated with a large class of weights, and give local results on the expected number of real zeros. We also show that the counting measures of properly scaled zeros of these random polynomials converge weakly to either the Ullman distribution or the arcsine distribution.
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