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L p Christoffel functions, L p universality, and Paley-Wiener spaces

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Abstract

Let ω be a regular measure on the unit circle in ℂ, and let p > 0. We establish asymptotic behavior, as n→∞, for the L p Christoffel function

$${\lambda _{n,p}}(\omega ,z) = \mathop {\inf }\limits_{\deg (P) \leqslant n - 1} \frac{{\int_{ - \pi }^\pi {{{\left| {P({e^{i\theta }})} \right|}^p}dw(\theta )} }}{{{{\left| {P(z)} \right|}^p}}}$$

at Lebesgue points z on the unit circle in ℂ, where ω′ is lower semi-continuous. While bounds for these are classical, asymptotics have never been established for p ≠ 2. The limit involves an extremal problem in Paley-Wiener space. As a consequence, we deduce universality type limits for the extremal polynomials, which reduce to random-matrix limits involving the sinc kernel in the case p = 2. We also present analogous results for L p Christoffel functions on [−1, 1].

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Authors and Affiliations

  1. Mathematics Department, The Open University of Israel, P.O. BOX 808, Raanana, 43107, Israel

    Eli Levin

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Doron S. Lubinsky

Authors
  1. Eli Levin
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  2. Doron S. Lubinsky
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Correspondence to Eli Levin.

Additional information

Research supported by NSF grant DMS1001182 and US-Israel BSF grant 2008399.

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Levin, E., Lubinsky, D.S. L p Christoffel functions, L p universality, and Paley-Wiener spaces. JAMA 125, 243–283 (2015). https://doi.org/10.1007/s11854-015-0008-2

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  • DOI: https://doi.org/10.1007/s11854-015-0008-2

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