Finite Limit Series on Chebyshev Polynomials, Orthogonal on Uniform Nets

In the paper we construct new series, called finite limit series on Chebyshev (Hahn) polynomials $\tau^{\alpha,\beta}_n(x)=\tau^{\alpha,\beta}_n(x,N)$, orthogonal on uniform net $\{0,1,\ldots,N-1\}$. Their partial sums $S_n(f;x)$ equal in boundary points $x=0$ и $x=N-1$ with approximated function $f(x)$. Construction of finite limit series based on the passage to the limit with $\alpha\to-1$ of Fourier series $\sum\limits_{k=0}^{N-1}f_k^\alpha \tau_k^{\alpha,\alpha}(x,N)$ on Chebyshev (Hahn) polynomials $\tau_n^{\alpha,\alpha}(x,N)$, orthonormal on uniform net $\{0,1,\ldots,N-1\}$.