On a two-dimensional analogue of the Lebesgue function for Fourier-Chebyshov sums

Moscow State University of Civil Engineering (National Research University), 26, Yaroslavskoe Shosse, 129337, Moscow, Russia

^* Corresponding author: aligadzhi@yandex.ru

Abstract

This article considers the problem of approximating a function of two variables f(x,y) by Fourier sums over Chebyshev polynomials orthogonal on a discrete grid. The paper shows that if T^α,β_n(x, N) (α,β > -1, n = 0,1,2,...) are classical Chebyshev polynomials, orthogonal on a discrete grid, then the system of polynomials of two variables {Z^α,β_m,n(x, y)}^k_m,n=0 = {T^α,β_m(x, N)}^k_m,n=0, where k = m + n ≤ N −1 is orthogonal on the set Ω_{N × N} = {(x_i, y_j)}_i,j=1^{N - 1}. For an arbitrary function f(x,y) continuous on [−1,1]², partial Fourier Chebyshev sums S^α,β_m,n,N(f, x,y) are constructed over the system of polynomials τ^α,β_m,n,N(x,y) orthonormal on the grid Ω_{N × N} = {(x_i, y_j)}_i,j=1^{N - 1}. The task is to estimate the partial sum S^α,β_m,n,N(f, x,y) of the Fourier series of the function f(x,y) from the system of polynomials τ^α,β_m,n,N(x,y) from the function itself f ∈ C_{[-1, 1]²}, in the case when(x,y) ∈ [−1,1]×[−1,1], which in turn reduces to the problem of estimating the Lebesgue function W^α,β_m,n,N(x,y). The result of the work is a theorem in which it is proved that the order of the Lebesgue constants ‖S^α,β_m,n,N‖ of the indicated discrete sums under certain conditions is O((mn)^q+1/2), where q = max{α,β}. As a consequence of the obtained result, some approximation properties of discrete sums S^α,β_m,n,N(f, x,y) are considered.