Browsing by Author "Levesley, Jeremy"

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Interpolation of Exponential-Type Functions on a Uniform Grid by Shifts of a Basis Function
(American Institute of Mathematical Sciences, 2021) Sun, Xinping; Kushpel, Alexander; Levesley, Jeremy; Jarad, Fahd
Interpolation of Exponential-Type Functions on a Uniform Grid by Shifts of a Basis Function
(Amer inst Mathematical Sciences-aims, 2021) Jarad, Fahd; Kushpel, Alexander; Levesley, Jeremy; Sun, Xinping
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).
Citation - WoS: 5
Citation - Scopus: 5
Widths and Entropy of Sets of Smooth Functions on Compact Homogeneous Manifolds
(Tubitak Scientific & Technological Research Council Turkey, 2021) Levesley, Jeremy; Tas, Kenan; Kushpel, Alexander
We develop a general method to calculate entropy and n-widths of sets of smooth functions on an arbitrary\rcompact homogeneous Riemannian manifold Md\r. Our method is essentially based on a detailed study of geometric\rcharacteristics of norms induced by subspaces of harmonics on Md\r. This approach has been developed in the cycle\rof works [1, 2, 10–19]. The method’s possibilities are not confined to the statements proved but can be applied in\rstudying more general problems. As an application, we establish sharp orders of entropy and n-widths of Sobolev’s\rclasses Wγ\rp\r(\rMd\r)\rand their generalisations in Lq\r(\rMd\r)\rfor any 1 < p, q < ∞. In the case p, q = 1, ∞ sharp in the power\rscale estimates are presented.