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

Let\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar \Omega $$ \end{document}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 :\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar \Omega $$ \end{document}N+2m → ℝ by algebraic polynomials on the grid ΩN={0, 1, …,N−1}. On the basis of two systems of Chebyshev polynomials orthogonal on the sets ΩN+m and ΩN, respectively, we construct a linear operatorYn+2m, N=Yn+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 ΩN):1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$|f(x) - \mathcal{Y}_{n + 2m,N} (f,x)| \leqslant c(m)z\Theta _{N,m} (x)\left[ {\frac{{x + 1}}{N}\left( {1 - \frac{x}{N}} \right)} \right]^{m/2 - 1/4} \frac{{E_{n + m[g,\ell _2 (\Omega _{N + m} )]} }}{{n^{m - 1/2} }}$$ \end{document} whereEn+m[g,l2(ΩN+m)] is the best approximation of the function1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$g(x) = g(x,m,N) = ((N - 1 + m)/2)^m \Delta ^{^m } f(x - m)$$ \end{document} by algebraic polynomials of degree at mostn+m in the spacel2 (ΩN+m) and the function ΘN, α(x) depends only on the weighted estimate for the Chebyshev polynomialsτkα,α(x, N).