Interpolation of exponential-type functions on a uniform grid by shifts of a basis function

Jeremy Levesley; Xinping Sun; Fahd Jarad; Alexander Kushpel

doi:10.3934/dcdss.2020403

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2021, Volume 14, Issue 7: 2399-2416. Doi: 10.3934/dcdss.2020403

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Interpolation of exponential-type functions on a uniform grid by shifts of a basis function

1.
Department of Mathematics, University of Leicester, Leicester, UK
2.
Department of Mathematics, Missouri State University, USA
3.
Department of Mathematics, Çankaya University, Ankara, Turkey

^* Corresponding author: Alexander Kushpel
^* Corresponding author: Alexander Kushpel

Dedicated to the memory of Ward Cheney

Received: November 30, 2019

Early access: June 2020

Published: July 2021

The first author was supported by EPSRC Grant EP/H020071/1, the University of Leicester via study and Missouri State University through a generous travel grant

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Abstract

In this paper, we present a new approach to solving the problem of interpolating a continuous function at $ (n+1) $ equally-spaced points in the interval $ [0, 1] $, using shifts of a kernel on the $ (1/n) $-spaced infinite grid. The archetypal example here is approximation using shifts of a Gaussian kernel. We present new results concerning interpolation of functions of exponential type, in particular, polynomials on the integer grid as a step en route to solve the general interpolation problem. For the Gaussian kernel we introduce a new class of polynomials, closely related to the probabilistic Hermite polynomials and show that evaluations of the polynomials at the integer points provide the coefficients of the interpolants. Finally we give a closed formula for the Gaussian interpolant of a continuous function on a uniform grid in the unit interval (assuming knowledge of the discrete moments of the Gaussian).

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Mathematics Subject Classification: Primary: 30E05; Secondary: 33C45.

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