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Christoffel Functions for Weights with Jumps

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Abstract

The asymptotic behavior of Christoffel functions is determined at points where the density of the corresponding measure has a jump discontinuity.

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Notes

  1. In this paper it will be \(\mathbb {R}\) but it could be \(\mathbb {C}\) or even a general abstract Hilbert space setting.

  2. However, A. Avila, Y. Last, and B. Simon’s 2010 paper [4] does have a.e.-type convergence theorems, see [4, Theorems 4, 5, p. 88].

  3. Following the tradition, if \(\mu \) is absolutely continuous with density (aka weight), say \(w\), then we write \(\lambda _n(w)\), \(p_n(w)\), and so forth. We will use Greek letters for measures and Latin ones for weights.

  4. When all energies are infinite, then the set \(S\) is called polar, and it does not have an equilibrium measure.

  5. Here and similarly below we use quotation marks to indicate that it is just an awkward way of forecasting that eventually we will take the infimum of some expression over all \(\eta >0\).

  6. \(\lfloor \cdot \rfloor \) denotes the mathematical integer part.

  7. As Walsh’s historical notes explain, it might be more appropriate to reference it as the Bernstein-M. Riesz-Walsh Lemma, see [25, p. 78]. For more unexpected revelations, see [13].

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Acknowledgments

We thank our friend Andrei Martínez-Finkelshtein for double checking the computations in [7] so we could use (3) that not only played a critical role in our proof, but, in fact, the entire inspiration for this paper came from reading [7] and from trying to combine it with ideas found in [11] to extend (3) to more general weights and measures. As it turned out, the right approach required tools that were way beyond the machinery introduced and developed in [11]. We also thank the referees for their valuable suggestions. The research of Paul Nevai was supported by KAU grant No. 20-130/1433 HiCi. The research of Vilmos Totik was supported by NSF DMS-1265375

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Authors and Affiliations

  1. KAU, Jeddah, Saudi Arabia

    Paul Nevai

  2. Upper Arlington (Columbus), Ohio, USA

    Paul Nevai

  3. MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Aradi v. tere 1, Szeged, 6720, Hungary

    Vilmos Totik

  4. Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave, CMC342, Tampa, FL, 33620-5700, USA

    Vilmos Totik

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  1. Paul Nevai
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Correspondence to Paul Nevai.

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Communicated by Arno Kuijlaars.

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Nevai, P., Totik, V. Christoffel Functions for Weights with Jumps. Constr Approx 42, 265–280 (2015). https://doi.org/10.1007/s00365-014-9255-1

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