On interpolation of functions of several variables

V. N. Chubarikov ; M. L. Sharapova

Čebyševskij sbornik, Tome 18 (2017) no. 4, pp. 339-347

Résumé

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 $f(x_1,\dots ,x_n),$ 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 $$ f(x_1,\dots ,x_n)=\sum_{m_1=-\infty}^{\infty}\dots \sum_{m_n=-\infty}^{\infty}c(m_1,\dots ,m_n)e^{2\pi i(m_1x_1+\dots +m_nx_n)}, $$ where $$ c(m_1,\dots ,m_n)=\int\limits_0^1\dots\int\limits_0^1f(x_1,\dots,x_n)e^{-2\pi i(m_1x_1+\dots +m_nx_n)}\;dx_1\dots dx_n. $$ Further, we select the number of lattice points $N$ in the form $N=N_1\dots N_n,$ 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 $$ P(x_1,\dots ,x_n)=\sum_{m_1=0}^{N_1-1}\dots\sum_{m_n=0}^{N_n-1}\tilde c(m_1,\dots ,m_n)e^{2\pi i(m_1x_1+\dots m_nx_n)}, $$ where $$ c(m_1,\dots ,m_n)=\frac 1N\sum_{k_1=1}^{N_1}\dots \sum_{k_n=1}^{N_n}f\left(\frac{M_1^{*}k_1}{N_1},\dots ,\frac{M_n^{*}k_n}{N_n}\right)e^{-2\pi i\left(\frac{M_1^{*}m_1}{N_1}+\dots+\frac{M_n^{*}m_n}{N_n}\right)}, $$ moreover $N_sM_s=N, M_sM_s^{*}\equiv 1\pmod{N_s}.$

Détail

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

Bibliographie

[1] Vinogradov I. M., Method of trigonometric sums in Number Theory, 2-nd edition, Nauka, M., 1980, 144 pp.

[2] Krylov A. N., Lectures on numerical calculations, ch. III, Gostehizdat, M., 1950 (in Russian)

[3] Babenko K. I., Founndations of numerical analysis, Nauka, M., 1986, 744 pp. (in Russian)

[4] Bahvalov N. S., Gidkov N. P., Kobel'kov G. M., Numerical methods, Text-book, Nauka, M., 1987, 600 pp. (in Russian)

[5] Korobov N. M., Number theoretical methods in numerical analysis, MZNMO, 2004, 288 pp. (in Russian)

[6] Hua L.-K., Selected Papers, Springer Verlag, New York, 1983, 888 pp. | MR | Zbl

[7] Wang Y., Selected Papers, Bejing, 1999, 458 pp.

[8] Voronin S. M., Selected Papers, Publ. N.E. Bauman MGTU, M., 2006, 480 pp. (in Russian)

[9] Arkhipov G. I., Selected Papers, Publ. Orël State Univ., Orel, 2013, 464 pp. (in Russian)

[10] Arkhipov G. I., Chubarikov V. N., Karatsuba A. A., Trigonometric Sums in Number Theory and Analysis, De Gruyter expositions in mathematics, 39, Berlin–New York, 2004, 554 pp. | MR | Zbl

[11] Ryaben'kii V. S., “On tables and interpolation of functions from some class”, DAN SSSR, 131:5 (1960), 1025–1027 | Zbl

[12] Chubarikov V. N., “Arithmetical sums from polynomial values”, Doklady RAS, 466:2 (2016), 152–153 | DOI | Zbl

[13] Chubarikov V. N., Scharapova M. L., “On a cubature formulae for periodic functions”, Bull. Moscow Univ. Ser.I, Math, mech., 2017, no. 6, 59–62

[14] Chubarikov V. N., Scharapova M. L., “On analogue of Gaussian quadrature for periodic functions”, Bull. of Cybernetics, 28:2 (2017), 60–65 (in Russian)

[15] Arkhipov G. I., Sadovnichii V. A., Chubarikov V. N., Lectures on mathematical analysis, University text-book, 4-th edition, Drofa, M., 2004, 640 pp. (in Russian)