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Notes on lacunary Müntz polynomials

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Abstract

We prove that a Müntz system has Chebyshev polynomials on [0,1] with uniformly bounded coefficients if and only if it is lacunary. A sharp Bernstein-type inequality for lacunary Müntz systems is established as well. As an application we show that a lacunary Müntz system fails to be dense inC(A) in the uniform norm for everyA ⊂ [0,1] with positive outer Lebesgue measure. A bounded Remez-type inequality is conjectured for non-dense Müntz systems on [0,1] which would solve Newman’s problem concerning the density of products of Müntz systems.

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Authors and Affiliations

  1. Department of Mathematics, Dalhousie University, B3H 3J5, Halifax, Nova Scotia, Canada

    Peter Borwein

  2. Department of Mathematics, The Ohio State University, 43210, Columbus, OH, USA

    Tamás Erdélyi

Authors
  1. Peter Borwein
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  2. Tamás Erdélyi
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Borwein, P., Erdélyi, T. Notes on lacunary Müntz polynomials. Israel J. Math. 76, 183–192 (1991). https://doi.org/10.1007/BF02782851

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  • DOI: https://doi.org/10.1007/BF02782851

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