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

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 Tα,βn(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 {Zα,βm,n(x, y)}km,n=0 = {Tα,βm(x, N)}km,n=0, where k = m + n ≤ N −1 is orthogonal on the set ΩN × N = {(xi, yj)}i,j=1N - 1. For an arbitrary function f(x,y) continuous on [−1,1]2, partial Fourier Chebyshev sums Sα,βm,n,N(f, x,y) are constructed over the system of polynomials τα,βm,n,N(x,y) orthonormal on the grid ΩN × N = {(xi, yj)}i,j=1N - 1. The task is to estimate the partial sum Sα,βm,n,N(f, x,y) of the Fourier series of the function f(x,y) from the system of polynomials τα,βm,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 Wα,βm,n,N(x,y). The result of the work is a theorem in which it is proved that the order of the Lebesgue constants ‖Sα,βm,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 Sα,βm,n,N(f, x,y) are considered.