About asymptotics of Chebyshev polynomials orthogonal on an uniform net

In this article asymptotic properties of the Chebyshev polynomials $T_n(x,N)$ ($0\le n\le N-1$) orthogonal on an uniform net $\Omega_N=\{0,1,\dots,N-1\}$ with the constant weight $\mu(x)=\frac2N$ (discrete analog of the Legendre polynomials) by $n=O(N^{\frac12})$, $N\to\infty$ were researched. The asymptotic formula that is relating polynomials $T_n(x,N)$ with Legendre polynomials $Pn(t)$ for $x=\frac N2(1+t)-\frac12$ was determined. The uniform estimation of remainder term of the formula relative to $t\in[-1,1]$, that in turn allows to prove unimprovable estimation of Chebyshev polynomials $T_n(x,N)$, was obtained.