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Approximations of Conjugate Functions by Partial Sums of Conjugate Fourier Series with Respect to a Certain System of Chebyshev – Markov Algebraic Fractions

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Abstract

In this paper, we study approximative properties of partial sums of a conjugate Fourier series with respect to a certain system of Chebyshev – Markov algebraic fractions. We cite the main results obtained in known works devoted to studying approximations of conjugate functions in polynomial and rational cases. We introduce a system of Chebyshev – Markov algebraic fractions and construct the corresponding conjugate rational Fourier – Chebyshev series. We obtain an integral representation for approximations of the conjugate function by partial sums of the constructed conjugate series. Moreover, we study approximations of the function, which is conjugate to \(|x|^s, 1 < s < 2,\) on the segment \([-1,1],\) by partial sums of the conjugate rational Fourier – Chebyshev series. We obtain an integral representation of approximations, establish their estimates, using the considered method, in dependence of the location of the point x on the segment, and find their asymptotic forms with \(n \to \infty\). We also calculate the optimal value of the parameter that makes deviations of partial sums of the conjugate rational Fourier – Chebyshev series from the conjugate function \(|x|^s, 1 < s < 2,\) tend to zero on the segment \([-1,1]\) at the highest possible rate. The obtained results have allowed us to thoroughly study properties of approximations of the function conjugate to \(|x|^s, s > 1,\) by partial sums of the conjugate Fourier series with respect to a system of Chebyshev polynomials of the first kind.

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REFERENCES

  1. Motornyi, V.P. “Asymptotically Exact Estimates of the Pointwise Approximation by Algebraic Polynomials of Certain Classes of Functions”, Dokl. Ross. Akad. Nauk 370 (3), 313–315 (2000).

  2. Motornyi, V.P. “On Asymptotically Exact Estimates of the Approximation by Algebraic Polynomials of Certain Classes of Functions”, Ukr. Matem. Zhurn. 52 (1), 85–99 (2000).

  3. Motornyi, V.P. “Approximation of a Class of Singular Integrals by Algebraic Polynomials with Regard to the Location of a Point on an Interval”, Trudy MiAN 232, 268–285 (2001).

  4. Korneichuk, N.P., Polovina, A.I. “On Approximation of Functions Satisfying a Lipschitz Condition by Algebraic Polynomials”, Matem. Zametki. 9 (4), 441–447 (1971).

  5. Privalov, I.I. “Sur les Fonctions Conjuguees”, Bull. Soc. Math. France 44, 100–103 (1916).

  6. Privalov, I.I. “To the Theory of Conjugate Trigonometric Series”, Matem. Sborn. 31 (2), 224–228 (1923).

  7. Kolmogorov, A.N. “Sur les Fonctions Harmoniques Conjuguées et les Séries de Fourier”, Fund. Math. 7, 24–29 (1925).

  8. Riesz, M. “Les Fonctions Conjuguées et les Séries de Fourier”, C. R. Acad. Sci. Paris 178, 1464–1467 (1924).

  9. Riesz, M. “Sur les Fonctions Conjuguées”, Math. Z. 27, 218–244 (1927).

  10. Bari, N.K. “On the Best Approximation of Two Conjugate Functions by Trigonometric Polynomials”, Izv. Akad. Nauk SSSR. Ser. Matem. 19 (5), 285–302 (1955).

  11. Stechkin, S.B. “On Best Approximation of Conjugate Functions by Trigonometric Polynomials”, Izv. Akad. Nauk SSSR. Ser. Matem. 20 (2), 197–206 (1956).

  12. Mardvilko, T.S., Pekarskii, A.A. “Conjugate Functions on the Closed Interval and Their Relationship with Uniform Rational and Piecewise Polynomial Approximations”, Math. Notes 99 (2), 272–283 (2016).

  13. Qureshi, K. “On the Degree of Approximation of Function Belonging to the Lipschitz Class by Means of a Conjugate Series”, Indian J. Pure Appl. Math. 12 (9), 1120–1123 (1981).

  14. Singh, U. “Approximation of Conjugate of Functions Belonging to Weighted Lipschitz Class \(W(L^p,\xi(t))\) by Hausdorff Means of Conjugate Fourier Series”, J. Comput. Appl. Math. 259 633–640 (2014).

  15. Rhoades, B.E. “The Degree of Approximation of Functions, and Their Conjugates, Belonging to Several General Lipschitz Classes by Hausdorff Matrix Means of Fourier series and Conjugate Series of a Fourier Series”, Tamkang of math. 45 (4), 389–395 (2014).

  16. Nigam, H.K., Sharma, A. “On Approximation of Conjugate of Functions Belonging to Different Classes by Product Means”, Int. J. Pure and Appl. Math. 76 (2), 303–316 (2012).

  17. Falaleev, L.P. “Approximation of Conjugate Functions by Cesáro Sums”, Matem. Zametki 28 (3), 451–458 (1980).

  18. Rusak, V.N., Rybachenko, I.V. “Uniform Rational Approximation of Conjugate Functions”, Vestn. BGU. Ser. 1. Matem. i Informatika 3, 83–86 (2013).

  19. Pekarskii, A.A. “Conjugate Functions and Their Connection with Uniform Rational and Piecewise-Polynomial Approximations”, Matem. Sborn. 206 (2), 175–182 (2015).

  20. Lukashov, A.L. “Chebyshev – Markov Algebraic Fractions on Several Segments”, Anal. Math. 24 (1), 111–130 (1998).

  21. Rouba, E.A., Patseika, P. “On One System of Rational Chebyshev – Markov Fractions”, Dokl. Nats. Akad. Nauk Belarusi 61 (1), 24–29 (2017).

  22. Rouba, Y., Patseika, P., Smatrytski, K. “On One System of Rational Chebyshev – Markov Fractions”, Anal. Math. 44 (1), 115–140 (2018).

  23. Vinogradov, I.M. Encyclopedia of Mathematics (in 5 volumes) (Sovetskaya Entsiklopediya, 1977).

  24. Evgrafov, M.A. Asymptotic Estimates and Entire Functions (Nauka, Moscow, 1979) [in Russian].

  25. Fedoryuk, M.V. Asymptotics of Integrals and Series (Nauka, Moscow, 1987) [in Russian].

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

  27. Rouba, E.A., Mikulich, E.G. “Constants in the Approximation of the Function \(|x|\) by Rational Interpolation Processes”, Dokl. Nats. Akad. Nauk Belarusi 53 (6), 11–15 (2009).

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Authors and Affiliations

  1. Yanka Kupala State University of Grodno, 22 Ozheshko str., 230023, Grodno, Republic of Belarus

    E. A. Rovba & P. G. Patseika

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  1. E. A. Rovba
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  2. P. G. Patseika
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Correspondence to E. A. Rovba or P. G. Patseika.

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Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 9, pp. 68–84.

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Rovba, E.A., Patseika, P.G. Approximations of Conjugate Functions by Partial Sums of Conjugate Fourier Series with Respect to a Certain System of Chebyshev – Markov Algebraic Fractions. Russ Math. 64, 61–75 (2020). https://doi.org/10.3103/S1066369X20090066

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  • DOI: https://doi.org/10.3103/S1066369X20090066

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