Geodesic
Finite Limit Series on Chebyshev Polynomials, Orthogonal on Uniform Nets
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 1, pp. 104-108.

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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 $\{0,1,\ldots,N-1\}$. 
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T. I. Sharapudinov. Finite Limit Series on Chebyshev Polynomials, Orthogonal on Uniform Nets. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 1, pp. 104-108. http://geodesic.mathdoc.fr/item/ISU_2013_13_1_a25/

[1] Sharapudinov I. I., Mixed series of orthogonal polynomials. Theory and applications, Dagestan. nauch. center RAN, Makhachkala, 2004, 276 pp.