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On certain integral operators associated with Hermite and Laguerre polynomials

Volume 45 / 2018

Mirosław Pawlak, Ulrich Stadtmüller Applicationes Mathematicae 45 (2018), 71-90 MSC: Primary 42C05; Secondary 62G07, 33C45. DOI: 10.4064/am2334-9-2017 Published online: 26 January 2018

Abstract

Motivated by applications in nonparametric curve estimation theory, we consider orthogonal polynomial representations for approximation of functions. In particular, we look at the classical Hermite and Laguerre polynomials that have been used in nonparametric statistics where instead of the complete knowledge of a function only its noisy samples are available. In order to study the consistency of the corresponding nonparametric series function estimators one should examine the first, second and often higher moments of estimators. These evaluations are related to higher powers of the associated kernel function of the selected approximating expansion. Hence, in this paper we investigate the asymptotic behavior of kernel operators related to powers of kernels resulting from such orthogonal polynomial expansions. We examine the case of kernel operators associated with Hermite and Laguerre polynomials.

Authors

  • Mirosław PawlakDepartment of Electrical
    and Computer Engineering
    University of Manitoba
    Winnipeg, Manitoba, Canada R3T 2N2
    e-mail
  • Ulrich StadtmüllerDepartment of Number Theory
    and Probability Theory
    Ulm University
    89069 Ulm, Germany
    e-mail

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