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 Chebyshevskii Sb., 2017, Volume 18, Issue 4, Pages 339–347 (Mi cheb616)

On interpolation of functions of several variables

V. N. Chubarikov, M. L. Sharapova

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: In this paper we constructed effective multivariate interpolation formulas for periodic functions, which are the precise on the Fourier polynomial classes. This paper continues investigations by N.M. Korobov [5], V.S. Rjaben'kii [11], S.M. Voronin [8], and others scientists on the application of the number-theoretic methods in numerical analysis. These authors was given the number of knots of a network equals to a prime number in the ring of integer rational numbers and in rings of integer numbers in algebraic numbers.
Here we consider the class of strictly regular periodic functions $f(x_1,…,x_n),$ having the period on of one the each variables, and expanding in the absolute convergent Fourier series (see, for example, [15], p. 447) of the form
$$f(x_1,…,x_n)=\sum_{m_1=-\infty}^{\infty}…\sum_{m_n=-\infty}^{\infty}c(m_1,…,m_n)e^{2\pi i(m_1x_1+…+m_nx_n)},$$
where
$$c(m_1,…,m_n)=\int\limits_0^1…\int\limits_0^1f(x_1,…,x_n)e^{-2\pi i(m_1x_1+…+m_nx_n)}\;dx_1…dx_n.$$
Further, we select the number of lattice points $N$ in the form $N=N_1…N_n,$ where $(N_s,N_t)=1$ as $s\ne t, 1\leq s,t\leq n,$ and $N_s\asymp N^{1/n}, 1\leq n,$ and using the Chinesse theorem on remainders, we construct the interpolation polynomial of the form
$$P(x_1,…,x_n)=\sum_{m_1=0}^{N_1-1}…\sum_{m_n=0}^{N_n-1}\tilde c(m_1,…,m_n)e^{2\pi i(m_1x_1+…m_nx_n)},$$
where
$$c(m_1,…,m_n)=\frac 1N\sum_{k_1=1}^{N_1}…\sum_{k_n=1}^{N_n}f(\frac{M_1^{*}k_1}{N_1},…,\frac{M_n^{*}k_n}{N_n})e^{-2\pi i(\frac{M_1^{*}m_1}{N_1}+…+\frac{M_n^{*}m_n}{N_n})},$$
moreover $N_sM_s=N, M_sM_s^{*}\equiv 1\pmod{N_s}.$

Keywords: the number-theoretic method in the numerical analysis, a lattice points, the V.S.Rjaben'kii method, the interpolation polynomial, rings of the integer rational and the integer algebraic numbers, the Chinesse theorem on remainders.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00071_à

DOI: https://doi.org/10.22405/2226-8383-2017-18-4-338-346

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UDC: 511.3
Revised: 11.12.2017
Accepted:14.12.2017

Citation: V. N. Chubarikov, M. L. Sharapova, “On interpolation of functions of several variables”, Chebyshevskii Sb., 18:4 (2017), 339–347

Citation in format AMSBIB
\Bibitem{ChuSha17} \by V.~N.~Chubarikov, M.~L.~Sharapova \paper On interpolation of functions of several variables \jour Chebyshevskii Sb. \yr 2017 \vol 18 \issue 4 \pages 339--347 \mathnet{http://mi.mathnet.ru/cheb616} \crossref{https://doi.org/10.22405/2226-8383-2017-18-4-338-346}