On interpolation of functions of several variables

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 $N_sM_s=N, M_sM_s^{*}\equiv 1\pmod{N_s}.$