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Asymptotic formulas for Lebesgue functions corresponding to the family of Lagrange interpolation polynomials

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Abstract

The asymptotic behavior of Lebesgue functions of trigonometric Lagrange interpolation polynomials constructed on an even number of nodes is studied. For these functions, asymptotic formulas involving concrete simplest trigonometric and algebraic-trigonometric polynomials were first obtained.

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References

  1. V. L. Goncharov, Theory of Interpolation and Approximations of Functions (GTTI, Moscow-Leningrad, 1934) [in Russian].

    Google Scholar 

  2. N. I. Akhiezer, Lectures in Approximation Theory (Gostekhizdat, Moscow-Leningrad, 1947) [in Russian].

    MATH  Google Scholar 

  3. I. P. Natanson, Constructive Theory of Functions (Gostekhizdat, Moscow-Leningrad, 1949) [in Russian].

    Google Scholar 

  4. P. P. Korovkin, Linear Operators and Approximation Theory (Fizmatgiz, Moscow, 1959) [in Russian].

    MATH  Google Scholar 

  5. A. F. Timan, Theory of Approximation of Functions of a Real Variable (Fizmatgiz, Moscow, 1960) [in Russian].

    Google Scholar 

  6. S. M. Nikol’skii, Approximation of Functions of Several Variables and Embedding Theorems (Nauka, Moscow, 1969) [in Russian].

    MATH  Google Scholar 

  7. V. M. Tikhomirov, Some Questions of Approximation Theory (Izd. Moskov. Univ., Moscow, 1976) [in Russian].

    Google Scholar 

  8. N. P. Korneichuk, Extremal Problems of Approximation Theory (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  9. V. K. Dzyadyk, Approximation Methods for the Solution of Differential and Integral Equations (Naukova Dumka, Kiev, 1988) [in Russian].

    MATH  Google Scholar 

  10. A. A. Privalov, Theory of Interpolation of Functions (Izd. Saratovsk. Univ., Saratov, 1990), Books 1–2 [in Russian].

    MATH  Google Scholar 

  11. K. I. Babenko, Foundations of Numerical Analysis (NITs “RKhD”, Moscow–Izhevsk, 2002) [in Russian].

    Google Scholar 

  12. I. A. Shakirov, “The Lagrange trigonometric interpolation polynomial with the minimal norm considered as an operator from C 2π to C 2π ” Izv. Vyssh. Uchebn. Zaved. Mat., No. 10, 60–68 (2010) [Russian Math. (Iz. VUZ) 54 (10), 52–59 (2010)].

    Google Scholar 

  13. A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 1960; Mir, Moscow, 1965), Vol. 2.

    MATH  Google Scholar 

  14. I. A. Shakirov, “A complete description of the Lebesgue functions for classical Lagrange interpolation polynomials,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 10, 80–88 (2011) [Russian Math. (Iz. VUZ) 55 (10), 70–77 (2011)].

    MATH  Google Scholar 

  15. I. A. Shakirov, “Lebesgue functions corresponding to a family of Lagrange interpolation polynomials,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 7, 77–89 (2013) [Russian Math. (Iz. VUZ) 57 (7), 66–76 (2013)].

    MATH  MathSciNet  Google Scholar 

  16. I. A. Shakirov, “On the fundamental characteristics of the family of Lagrange interpolation polynomials,” Izv. Saratovsk. Univ. Ser. Mat. Mekh. Inform. 13 (1 (2)), 99–104 (2013).

    MATH  Google Scholar 

  17. T. Rivlin, “The Lebesgue constants for polynomial interpolation,” in Functional Analysis and Its Application, Ed. by H. C. Garnier et al. (Springer-Verlag, Berlin, 1974), pp. 422–437.

    Chapter  Google Scholar 

  18. L. Brutman, “Lebesgue functions for polynomial interpolation is a survey,” Ann. Numer. Math. 4, 111–127 (1997).

    MATH  MathSciNet  Google Scholar 

  19. P. Vertesi, “Optimal Lebesgue constants for Lagrange interpolation,” SIAM J. Numer. Anal. 27, 1322–1331 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  20. R. Gunttner, “Evaluation Lebesgue constant,” SIAM J. Numer. Anal. 17 (4), 512–520 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  21. I. A. Shakirov, “Influence of the choice of Lagrange interpolation nodes on the exact and approximate values of the Lebesgue constants,” Sibirsk. Mat. Zh. 55 (6), 1404–1423 (2014) [Sib. Math. J. 55 (6), 1144–1160 (2014)].

    MATH  MathSciNet  Google Scholar 

  22. I. A. Shakirov, “Quadrature formulas for a singular integral with shift and their applications,” Differ. Uravn. 27 (4), 682–691 (1991) [Differ. Equations 27 (4), 487–494 (1991)].

    MATH  MathSciNet  Google Scholar 

  23. I. A. Shakirov, “Exact values of the norms on L 2[0, 2π] of the Lagrange polynomials defined on an even number of nodes,” Izv. Vyssh. Uchebn. Zaved. Mat. Severn. Kavkaz. Region. Estestv. Nauki Spets. Vypusk, 77–79 (2011).

    Google Scholar 

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Authors and Affiliations

  1. Naberezhnye Chelny State Pedagogical University, Naberezhnye Chelny, Russia

    I. A. Shakirov

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  1. I. A. Shakirov
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Correspondence to I. A. Shakirov.

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Original Russian Text © I. A. Shakirov, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 1, pp. 133–147.

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Shakirov, I.A. Asymptotic formulas for Lebesgue functions corresponding to the family of Lagrange interpolation polynomials. Math Notes 102, 111–123 (2017). https://doi.org/10.1134/S0001434617070124

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