Approximation of discrete functions and Chebyshev polynomials orthogonal on the uniform grid

Abstract

Let$\bar \Omega$ _N+2m ={−m, −m+1, …, −1, 0, 1, …,N−1,N, …,N−1+m}. The present paper is devoted to the approximation of discrete functions of the formf :$\bar \Omega$ _N+2m → ℝ by algebraic polynomials on the grid Ω _N ={0, 1, …,N−1}. On the basis of two systems of Chebyshev polynomials orthogonal on the sets Ω _N+m and Ω _N , respectively, we construct a linear operatorY _{n+2m, N} =Y _{n+2m, N} (f), acting in the space of discrete functions as an algebraic polynomial of degree at mostn+2m for which the following estimate holds (x ε Ω _N ):

$$|f(x) - \mathcal{Y}_{n + 2m,N} (f,x)| \leqslant c(m)z\Theta _{N,m} (x)\left[ {\frac{{x + 1}}{N}\left( {1 - \frac{x}{N}} \right)} \right]^{m/2 - 1/4} \frac{{E_{n + m[g,\ell _2 (\Omega _{N + m} )]} }}{{n^{m - 1/2} }}$$

(1)

whereE _n+m[g,l ₂(Ω _N+m )] is the best approximation of the function

$$g(x) = g(x,m,N) = ((N - 1 + m)/2)^m \Delta ^{^m } f(x - m)$$

(1)

by algebraic polynomials of degree at mostn+m in the spacel ₂ (Ω _N+m ) and the function Θ _{N, α} (x) depends only on the weighted estimate for the Chebyshev polynomialsτ ^α,α_k (x, N).