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1. Local Asymptotics for Orthonormal Polynomials on the Unit Circle Via Universality

   Lubinsky, D. (January 2020, Journal d'Analyse Mathematique (it won't recognize the journal))

   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(ez 

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2. The effect of adding mass points on bounds for orthogonal polynomials

   Lubinsky, D. (January 2021, Dolomites Research Notes on Approximation)

   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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3. Pointwise Asymptotics for Orthonormal Polynomials at the Endpoints of the Interval Via Universality

   Lubinsky, Doron (January 2019, International mathematics research notices)

   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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4. ALGEBRAIC STRUCTURE OF THE RANGE OF A TRIGONOMETRIC POLYNOMIAL

   https://doi.org/10.1017/S0004972719001229  

   KOVALEV, LEONID V.; YANG, XUERUI (October 2020, Bulletin of the Australian Mathematical Society)

   null (Ed.)

   The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with a certain symmetry. 

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5. On correlation bounds against polynomials Leibniz International Proceedings in Informatics (LIPIcs):38th Computational Complexity Conference (CCC 2023)

   https://doi.org/10.4230/LIPIcs.CCC.2023.3  

   Ivanov, Peter; Pavlovic, Liam; Viola, Emanuele (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Ta-Shma, Amnon (Ed.)

   We study the fundamental challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over 𝔽₂. Our main contributions include: 1) In STOC 2020, CHHLZ introduced a new technique to prove correlation bounds. Using their technique they established new correlation bounds for low-degree polynomials. They conjectured that their technique generalizes to higher degree polynomials as well. We give a counterexample to their conjecture, in fact ruling out weaker parameters and showing what they prove is essentially the best possible. 2) We propose a new approach for proving correlation bounds with the central "mod functions," consisting of two steps: (I) the polynomials that maximize correlation are symmetric and (II) symmetric polynomials have small correlation. Contrary to related results in the literature, we conjecture that (I) is true. We argue this approach is not affected by existing "barrier results." 3) We prove our conjecture for quadratic polynomials. Specifically, we determine the maximum possible correlation between quadratic polynomials modulo 2 and the functions (x_1,… ,x_n) → z^{∑ x_i} for any z on the complex unit circle, and show that it is achieved by symmetric polynomials. To obtain our results we develop a new proof technique: we express correlation in terms of directional derivatives and analyze it by slowly restricting the direction. 4) We make partial progress on the conjecture for cubic polynomials, in particular proving tight correlation bounds for cubic polynomials whose degree-3 part is symmetric. 

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