# Mixed Series of Chebyshev Polynomials Orthogonal on a Uniform Grid

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## Abstract

We construct an expansion of a discrete function in the form of a mixed series of Chebyshev polynomials. We obtain estimates of the approximation error of the function and its derivatives.

## Key words

Chebyshev polynomials mixed series of orthogonal polynomials approximation of discrete functions discrete Fourier-Chebyshev series Christoffel-Darboux formula## Preview

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## REFERENCES

- 1.I. I. Sharapudinov, “Approximations of functions with variable smoothness by Fourier sums over orthogonal polynomials,” in:
*International Conference “Approximation Theory and Harmonic Analysis”*(*May 26–29, 1998*) [in Russian], Tula, 1998, p. 275.Google Scholar - 2.I. I. Sharapudinov, “Approximation to discrete functions and Chebyshev polynomials orthogonal on the uniform grid,” in:
*Voronezh Winter Mathematical Workshop “Current Methods of Function Theory and Related Problems*(*January 27–February 4, 1999*)”, Abstracts of Papers [in Russian], Voronezh, 1999, p. 204.Google Scholar - 3.I. I. Sharapudinov, “Corrected Fourier sums over orthogonal polynomials and their approximation properties,” in:
*Voronezh Winter Mathematical Workshop “Current Methods of Function Theory and Related Problems*(*January 27–February 4, 2001*)”, Abstracts of Papers [in Russian], Voronezh, 2001, pp. 289–290.Google Scholar - 4.I. I. Sharapudinov, “Mixed series of orthogonal polynomials,” in:
*10th Saratov Winter Workshop “Current Problems of Function Theory and Their Applications*(*January 28–February 4, 2002*)”, Abstracts of Papers [in Russian], Saratov, 2002, pp. 228–229.Google Scholar - 5.I. I. Sharapudinov, “Mixed series of polynomials orthogonal on discrete grids,” in:
*Voronezh Winter Mathematical Workshop “Current Methods of Function Theory and Related Problems*(*January 26–February 2, 2003*)”, Abstracts of Papers [in Russian], Voronezh, 2003, pp. 285–286.Google Scholar - 6.I. I. Sharapudinov, “Mixed series of Jacobi polynomials and methods of their discretization in the Chebyshev case,” in:
*Sixth Kazan International Summer Workshop*(June 27–July 4, 2003),*Trudy Mat. Tsentra im. Lobachevskogo*, vol. 19 “Function Theory, Its Applications, and Related Topics” [in Russian], Saratov, Kazan, 2003.Google Scholar - 7.I. I. Sharapudinov, “Mixed series involving some orthogonal systems and their applications,” in:
*12th Saratov Winter Workshop “Current Problems of Function Theory and Their Applications*(*January 27–February 3, 2004*)”, Abstracts of Papers [in Russian], Saratov, 2004, pp. 205–206.Google Scholar - 8.I. I. Sharapudinov, “Mixed series of ultraspherical polynomials and their approximation properties,”
*Mat. Sb.*[*Russian Acad. Sci. Sb. Math.*],**194**(2003), no. 3, 115–148.Google Scholar - 9.I. I. Sharapudinov, “Approximation properties of the operators \(\mathcal{Y}_{n + 2r} (f)\) and their discrete analogs,”
*Mat. Zametki*[*Math. Notes*],**72**(2002), no. 5, 765–795.Google Scholar - 10.I. I. Sharapudinov,
*Polynomials Orthogonal on Discrete Grids*[in Russian], Izd. Dagestan. Gos. Ped. Inst. Makhachkala, 1997.Google Scholar - 11.S. Karlin and J. L. McGregor, “The Hahn polynomials, formulas, and an application,”
*Scripta Math.*,**26**(1961), no. 1, 33–46.Google Scholar - 12.G. Gasper, “Positivity and special functions,” in:
*Theory and Application of Special Functions, Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975*(R. A. Askey, editor), Math. Res. Center, Univ. Wisconsin, Publ. no. 35, Academic Press, New York, 1975, pp. 375–433.Google Scholar - 13.G. Szego,
*Orthogonal Polynomials*, Colloquium Publ., vol. XXIII, Amer. Math. Soc., Providence, RI, 1959; Russian translation: Fizmatgiz, Moscow, 1962.Google Scholar

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