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The Riesz–Zygmund Sums of Fourier–Chebyshev Rational Integral Operators and Their Approximation Properties

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Abstract

Studying the approximation properties of a certain Riesz–Zygmund sum of Fourier–Chebyshev rational integral operators with constraints on the number of geometrically distinct poles, we obtain an integral expression of the operators. We find upper bounds for pointwise and uniform approximations to the function \( |x|^{s} \) with \( s\in(0,2) \) on the segment \( [-1,1] \), an asymptotic expression for the majorant of uniform approximations, and the optimal values of the parameter of the approximant providing the greatest decrease rate of the majorant. We separately study the approximation properties of the Riesz–Zygmund sums for Fourier–Chebyshev polynomial series, establish an asymptotic expression for the Lebesgue constants, and estimate approximations to \( f\in H^{(\gamma)}[-1,1] \) and \( \gamma\in(0,1] \) as well as pointwise and uniform approximations to the function \( |x|^{s} \) with \( s\in(0,2) \).

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References

  1. Hardy G.H. and Riesz M., The General Theory of Dirichlet’s Series, Cambridge University, Cambridge (1915).

    Google Scholar 

  2. Zygmund A., “The approximation of functions by typical means of their Fourier series,” Duke Math. J., vol. 12, no. 4, 695–704 (1945).

    Article  MathSciNet  Google Scholar 

  3. Oberchoff N., “Applications de la sommation par les moyennes arithmétiques dans la théorie des séries de Fourier, des séries sphériques et ultrasphériques,” Bull. Math. Soc. Sci. Math. Roumanie (N.S.), vol. 40, no. 1, 27–38 (1938).

    Google Scholar 

  4. Kwee B., “The approximation of continuous functions by Riesz typical means of their Fourier series,” J. Austral. Math. Soc., vol. 7, no. 4, 539–544 (1967).

    MathSciNet  Google Scholar 

  5. Stepanyants S.A., “A problem of inclusion of discrete Riesz means methods,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., vol. 4, 12–17 (2007).

    Google Scholar 

  6. Hahinov I.V., “Interconnection of Cesàro methods and discrete Riesz means,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., vol. 5, 51–55 (2011).

    Google Scholar 

  7. Il’yasov N.A., “Approximation of periodic functions by Zygmund means,” Math. Notes, vol. 39, no. 3, 200–209 (1986).

    Article  MathSciNet  Google Scholar 

  8. Geit V.E., “Embedding theorems for Boas classes,” Russian Math. (Iz. VUZ), vol. 40, no. 5, 27–31 (1996).

    MathSciNet  Google Scholar 

  9. Stepanets A.I., “Approximate properties of the Zygmund method,” Ukrainian Math. J., vol. 51, no. 4, 493–518 (1999).

    Article  MathSciNet  Google Scholar 

  10. Chikina T.S., “Approximation by Zygmund–Riesz means in the \( p \)-variation metric,” Anal. Math., vol. 39, no. 1, 29–44 (2013).

    Article  MathSciNet  Google Scholar 

  11. Volosivets S.S. and Likhacheva T.B., “Several questions of approximation by polynomials with respect to multiplicative systems in weighted \( L_{p} \) spaces,” Izv. Saratov University. Math. Mech. Inform., vol. 15, no. 3, 251–258 (2015).

    Article  Google Scholar 

  12. Rusak V.N., “A method of approximation by rational functions,” Vestsi Akad. Navuk Belarusi Ser. Fiz.-Mat. Navuk, vol. 3, 15–20 (1978).

    Google Scholar 

  13. Rovba E.A., “Rational integral operators on a segment,” Vestnik BGU, vol. 1, no. 1, 34–39 (1996).

    MathSciNet  Google Scholar 

  14. Smotritskii K.A., “Approximation of convex functions by rational integral operators on a line,” Vestn. Belarus. Gos. University. Fiz., Mat., Inform., no. 3, 64–70 (2005).

    MathSciNet  Google Scholar 

  15. Rovba E.A. and Potseiko P.G., “Riesz–Zygmund means of rational Fourier–Chebyshev series and approximations of the function \( |x|^{s} \),” Tr. Inst. Mat., vol. 28, no. 1–2, 74–90 (2020).

    MathSciNet  Google Scholar 

  16. Rovba E.A., “On a direct method in a rational approximation,” Dokl. Nats. Akad. Nauk Belarusi, vol. 23, no. 11, 968–971 (1979).

    MathSciNet  Google Scholar 

  17. Potseiko P.G. and Rovba E.A., “Approximations on classes of Poisson integrals by Fourier–Chebyshev rational integral operators,” Sib. Math. J., vol. 62, no. 2, 292–312 (2021).

    Article  MathSciNet  Google Scholar 

  18. Potseiko P.G. and Rovba E.A., “Conjugate rational Fourier–Chebyshev operator and its approximation properties,” Russian Math. (Iz. VUZ), vol. 66, no. 3, 35–49 (2022).

    Article  MathSciNet  Google Scholar 

  19. Potseiko P.G., Rovba E.A., and Smotritskii K.A., “On one rational integral operator of Fourier–Chebyshev type and approximation of Markov functions,” J. Belarusian State University. Math. Inform., vol. 2, 6–27 (2020).

    Article  Google Scholar 

  20. Potseiko P.G. and Rovba E.A., “On rational Abel–Poisson means on a segment and approximations of Markov functions,” J. Belarusian State University. Math. Inform., vol. 3, 6–24 (2021).

    Article  MathSciNet  Google Scholar 

  21. Potseiko P.G. and Rovba E.A., “On rational approximations of the Markov function on the segment by the Fejer sums with a fixed number of poles,” Tr. Inst. Mat., vol. 30, no. 1, 57–77 (2022).

    Google Scholar 

  22. Bernstein S., “Sur meilleure approximation de \( |x| \) par des polynomés de degrés donnés,” Acta Math., vol. 37, no. 1, 1–57 (1914).

    Article  MathSciNet  Google Scholar 

  23. Newman D.J., “Rational approximation to \( |x| \),” Michigan Math. J., vol. 11, no. 1, 11–14 (1964).

    Article  MathSciNet  Google Scholar 

  24. Bulanov A.P., “Asymptotics for least deviation of \( |x| \) from rational functions,” Math. USSR-Sb., vol. 5, no. 2, 275–290 (1968).

    Article  Google Scholar 

  25. Vyacheslavov N.S., “Approximation of the function \( |x| \) by rational functions,” Math. Notes, vol. 16, no. 1, 680–685 (1974).

    Article  MathSciNet  Google Scholar 

  26. Stahl H., “Best uniform rational approximation of \( |x| \) on \( [-1,1] \),” Sb. Math., vol. 76, no. 2, 461–487 (1993).

    Article  MathSciNet  Google Scholar 

  27. Bernstein S., “Sur la meilleure approximation de \( |x|^{p} \) par des polynomés de degrés trés eléves,” Izv. Akad. Nauk SSSR Ser. Mat, vol. 2, no. 2, 169–190 (1938).

    Google Scholar 

  28. Freud G. and Szabados J., “Rational approximation to \( x^{\alpha} \),” Acta Math. Acad. Sci. Hungar., vol. 18, no. 3, 393–399 (1967).

    Article  MathSciNet  Google Scholar 

  29. Gonchar A.A., “On the rapidity of rational approximation of continuous functions with characteristic singularities,” Math. USSR-Sb., vol. 2, no. 4, 561–568 (1967).

    Article  Google Scholar 

  30. Vyacheslavov N.S., “On the approximation of \( x^{\alpha} \) by rational functions,” Math. USSR-Izv., vol. 16, no. 1, 83–101 (1981).

    Article  MathSciNet  Google Scholar 

  31. Stahl H., “Best uniform rational approximation of \( x^{\alpha} \) on \( [0,1] \),” Bull. Amer. Math. Soc., vol. 28, no. 1, 116–122 (1993).

    Article  MathSciNet  Google Scholar 

  32. Revers M., “On the asymptotics of polynomial interpolation to \( x^{\alpha} \) at the Chebyshev nodes,” J. Approx. Theory, vol. 165, no. 1, 70–82 (2013).

    Article  MathSciNet  Google Scholar 

  33. Ganzburg M.I., “The Bernstein constant and polynomial interpolation at the Chebyshev nodes,” J. Approx. Theory, vol. 119, no. 2, 193–213 (2002).

    Article  MathSciNet  Google Scholar 

  34. Raitsin R.A., “Asymptotic properties of uniform approximations of functions with algebraic singularities by partial sums of a Fourier–Chebyshev series,” Izv. Vyssh. Uchebn. Zaved. Matematika, vol. 3, 45–49 (1980).

    MathSciNet  Google Scholar 

  35. Lungu K.N., “On best approximations by rational functions with a fixed number of poles,” Sb. Math., vol. 15, no. 2, 313–324 (1971).

    Article  MathSciNet  Google Scholar 

  36. Lungu K.N., “Best approximations by rational functions with a fixed number of poles,” Sib. Math. J., vol. 25, no. 2, 289–296 (1984).

    Article  MathSciNet  Google Scholar 

  37. Stepanets A.I., Uniform Approximations by Trigonometric Polynomials, Utrecht, Boston, Koln, and Tokyo (2001).

    Book  Google Scholar 

  38. Bauer S.M., Filippov S.B., Smirnov A.L., Tovstik P.E., and Vaillancourt R., Asymptotic Methods in Mechanics of Solids, Birkhäuser, Basel (2015) (International Series of Numerical Mathematics; vol. 167).

    Book  Google Scholar 

  39. Sidorov Yu.V., Fedoryuk M.V., and Shabunin M.I., Lectures in the Theory of Functions of a Complex Variable, Nauka, Moscow (1989) [Russian].

    Google Scholar 

  40. Evgrafov M.A., Asymptotic Estimates and Entire Functions, Fizmatgiz, Moscow (1962) [Russian].

    Google Scholar 

  41. Fedoryuk M.V., Asymptotics: Integrals and Series, Nauka, Moscow (1987) [Russian].

    Google Scholar 

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Funding

The authors were supported by the Fundamental Research Program “Convergence 2020” (Grant no. 20162269).

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Authors and Affiliations

  1. Yanka Kupala State University of Grodno, Grodno, Belarus

    P. G. Potseiko & E. A. Rovba

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  1. P. G. Potseiko
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  2. E. A. Rovba
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Correspondence to P. G. Potseiko.

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Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2024, Vol. 65, No. 1, pp. 140–163. https://doi.org/10.33048/smzh.2024.65.112

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Potseiko, P.G., Rovba, E.A. The Riesz–Zygmund Sums of Fourier–Chebyshev Rational Integral Operators and Their Approximation Properties. Sib Math J 65, 118–138 (2024). https://doi.org/10.1134/S0037446624010129

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  • DOI: https://doi.org/10.1134/S0037446624010129

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