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Marcinkiewicz–Zygmund type results in multivariate domains

Abstract

We investigate Marcinkiewicz–Zygmund type inequalities for multivariate polynomials on various compact domains in \({\mathbb{R}^d}\). These inequalities provide a basic tool for the discretization of the L p norm and are widely used in the study of the convergence properties of Fourier series, interpolation processes and orthogonal expansions. Recently Marcinkiewicz–Zygmund type inequalities were verified for univariate polynomials for the general class of doubling weights, and for multivariate polynomials on the ball and sphere with doubling weights. The main goal of the present paper is to extend these considerations to more general multidimensional domains, which in particular include polytopes, cones, spherical sectors, toruses, etc. Our approach will rely on application of various polynomial inequalities, such as Bernstein–Markov, Schur and Videnskii type estimates, and also using symmetry and rotation in order to generate results on new domains.

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Correspondence to A. Kroó.

Additional information

This work has been partially supported by the BIRD163015 and DOR funds of the University of Padova. The second author was supported by the Visiting Professors program year 2017 of the Department of Mathematics “Tullio Levi-Civita” of the University of Padova and by the OTKA Grant K111742. This research has been accomplished within the RITA “Research ITalian network on Approximation”.

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De Marchi, S., Kroó, A. Marcinkiewicz–Zygmund type results in multivariate domains. Acta Math. Hungar. 154, 69–89 (2018). https://doi.org/10.1007/s10474-017-0769-4

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  • DOI: https://doi.org/10.1007/s10474-017-0769-4

Key words and phrases

  • multivariate polynomial
  • Marcinkiewicz–Zygmund type inequality
  • L p optimal mesh

Mathematics Subject Classification

  • 41A17
  • 41A63