ISSN 2304-0122 Ufa Mathematical Journal. Vol. 5. No 1 (2013). P. 90-101. doi:10.13108/2013-5-1-90 UDC 519.65 CONSTRUCTION OF OPTIMAL GRID INTERPOLATION FORMULAS IN SOBOLEV SPACE m 2 ( ) OF PERIODIC FUNCTION OF VARIABLES BY SOBOLEV METHOD N.H. MAMATOVA, A.R. HAYOTOV, KH.M. SHADIMETOV Abstract. In the present work we consider the problem of constructing optimal grid interpolation formulas in the space L 2 (H ) of periodic function of n variables. We find the coefficients of grid interpolation formulas. Keywords: Sobolev space, optimal interpolation formula, properties of the discrete ana- logue of the operator Δ , optimal coefficients. 1. Main results Let us remind the definition of Sobolev space () 2 ( ) of periodic functions of variables. Let a function () be periodic with the periods matrix , ( + )= (), ∈ R , where is an arbitrary integer column vector, is a matrix of size × having unit determi- nant. To the matrix we associate its fundamental parallelepiped Ω 0 letting Ω 0 = { ∈ R : = , 0 < 1, =1, 2,...,}. Suppose 2> and the integral ∫ Ω 0 ∑ ||= ! ! ( ()) 2 is finite, where is a multi-index, =( 1 , 2 ,..., ), != 1 ! 2 ! ... !, || = ∑ =1 , ()= |α| () α 1 1 α 2 2 ... αn n . The norm in () 2 ( ) is defined by the formula ‖()| () 2 ( )|| = ⎡ ⎣ ∫ Ω 0 ∑ ||= ! ! ( ()) 2 ⎤ ⎦ 1 2 . As the elements of the spaces () 2 ( ) serve the functions differing by a constant. The space ()* 2 ( ) comprises of all periodic functionals orthogonal to one, i.e., (ℓ(), 1) = 0. (1) N.H. Mamatova, A.R. Hayotov, Kh.M. Shadimetov, Construction of optimal grid interpola- tion formulas in Sobolev space L m 2 (H) of periodic function of n variables by Sobolev method. c ○ Mamatova N.H., Hayotov A.R., Shadimetov Kh.M. 2013. Submitted December 20, 2011. 90