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Estimates of the Values of \(n\)-Widths of Classes of Analytic Functions in the Weight Spaces \(H_{2,\gamma}(D)\)

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Abstract

In a simply connected bounded domain \(D\subset\mathbb C\) with rectifiable Jordan boundary \(\partial D\), we study the classes \(H_{2,\gamma}(D;\Omega_k,\Phi)\), \(k\in\mathbb N\), consisting of analytic functions \(f\in H_{2,\gamma}(D)\) in \(D\) each of which, for any \(t\in(0,1)\), satisfies the condition \(\Omega_k(f,t)\le\Phi(t)\). Here \(\Omega_k(f)\) is the generalized modulus of continuity of \(k\)th order in \(H_{2,\gamma}(D)\) and \(\Phi\) is a majorant. For these classes, we find upper and lower bounds for various \(n\)-widths, as well as upper bounds for the moduli of Fourier coefficients. We obtain a constraint on the majorant \(\Phi\) under which the exact values of these extremal characteristics can be calculated. In the case of the unit disk, similar results are obtained for classes of analytic functions whose definitions include the Hadamard compositions \(\mathscr D(\mathscr B_m,f)\) in addition to \(\Omega_k(f)\) and \(\Phi\). Concrete realizations of some obtained exact results are presented.

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References

  1. V. M. Tikhomirov, “Approximation theory,” in Analysis – 2, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. (VINITI, Moscow, 1987), Vol. 14, pp. 103–260.

    MATH  Google Scholar 

  2. A. Pinkus, \(n\)-Widths in Approximation Theory (Springer- Verlag, Berlin, 1985).

    Book  MATH  Google Scholar 

  3. S. D. Fisher, Function Theory on Planar Domains. A Second Course in Complex Analyses (John Wiley & Sons, New York, 2007).

    Google Scholar 

  4. Yu. A. Farkov, “On the best linear approximation of holomorphic functions,” J. Math. Sci. 218 (5), 678–698 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  5. S. B. Vakarchuk and M. Sh. Shabozov, “The widths of classes of analytic functions in a disc,” Sb. Math. 201 (8), 1091–1110 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  6. L. V. Taikov, “Diameters of certain classes of analytic functions,” Math. Notes 22 (2), 650–656 (1977).

  7. N. Ainulloev and L. V. Taikov, “Best approximation in the sense of Kolmogorov of classes of functions analytic in the unit disc,” Math. Notes 40 (3), 699–705 (1986).

    Article  MATH  Google Scholar 

  8. V. P. Zakharyuta and N. I. Skiba, “Estimates of \(n\)-diameters of some classes of functions analytic on Riemann surfaces,” Math. Notes 19 (6), 525–532 (1976).

    MATH  Google Scholar 

  9. M. Z. Dveirin, “Problems of best approximation of classes of functions analytic in the unit disk,” in Theory of Approximation of Functions (Nauka, Moscow, 1977), pp. 129–132 [in Russian].

    MathSciNet  Google Scholar 

  10. O. G. Parfenov, “Widths of a class of analytic functions,” Math. USSR-Sb. 45 (2), 283–289 (1983).

    Article  MATH  Google Scholar 

  11. Yu. A. Farkov, “Widths of Hardy classes and Bergman classes on the ball in \(\mathbb C^n\),” Russian Math. Surveys 45 (5), 229–231 (1990).

    MathSciNet  MATH  Google Scholar 

  12. K. Yu. Osipenko and M. I. Stesin, “On widths of the Hardy class \(H_n\) in the \(n\)-dimensional ball,” Russian Math. Surveys 45 (5), 235–236 (1990).

    MathSciNet  MATH  Google Scholar 

  13. S. B. Vakarchuk, “Exact values of widths of classes of analytic functions on the disk and best linear approximation methods,” Math. Notes 72 (5), 615–619 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  14. S. B. Vakarchuk and V. I. Zabutnaya, “Best linear approximation methods for functions of Taikov classes in the Hardy spaces \(H_{q,\rho}\), \(q\ge1\), \(0<\rho\le1\),” Math. Notes 85 (3), 322–327 (2009).

    MathSciNet  MATH  Google Scholar 

  15. M. Sh. Shabozov and G. A. Yusupov, “Best approximation methods and widths for some classes of functions in \(H_{q,\rho}\), \(1\le q\le\infty\), \(0<\rho\le1\),” Siberian Math. J. 57 (2), 369–376 (2016).

    MathSciNet  MATH  Google Scholar 

  16. V. A. Abilov, F. V. Abilova, and M. K. Kerimov, “Sharp estimates for the rate of convergence of Fourier series of functions of a complex variable in the space ,” Comput. Math. Math. Phys. 50 (6), 946–950 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Sh. Shabozov and M. S. Saidusaynov, “Upper bounds for the approximation of certain classes of functions of a complex variable by Fourier series in the space \(L_2\) and n-widths,” Math. Notes 103 (4), 656–668 (2018).

    MathSciNet  MATH  Google Scholar 

  18. Yu. I. Grigoryan, “Diameters of certain sets in function spaces,” Uspekhi Mat. Nauk 30 (3 (183)), 161–162 (1975).

    MathSciNet  Google Scholar 

  19. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Applications (Nauka i Tekhnika and the technique, Minsk, 1987) [in Russian].

    MATH  Google Scholar 

  20. J. T. Scheik, “Polynomial approximation of the functions analytic in a disk,” Proc. Amer. Math. Soc. 17 (6), 1238–1243 (1966).

    Article  MathSciNet  Google Scholar 

  21. S. B. Vakarchuk, “On estimates in \(L_2(\mathbb{R})\) of mean \(\nu\)-widths of classes of functions defined via the generalized modulus of continuity of \(\omega_{\mathcal{M}}\),” Math. Notes 106 (2), 191–202 (2019).

    MathSciNet  MATH  Google Scholar 

  22. S. B. Vakarchuk, “Approximation by classical orthogonal polynomials with weight in spaces \(L_{2,\gamma}(a,b)\) and widths of some functional classes,” Russian Math. (Iz. VUZ) 63 (12), 32–44 (2019).

    MathSciNet  MATH  Google Scholar 

  23. D. Gaier, Lectures on the Theory of Approximation in the Complex Domain (Mir, Moscow, 1986) [in Russian].

    MATH  Google Scholar 

  24. A. I. Markushevich, Theory of Analytic Functions. Vol. II: Further Construction of the Theory (Nauka, Moscow, 1968) [in Russian].

    Google Scholar 

  25. P. K. Suetin, “Polynomials orthogonal over a region and Bieberbach polynomials,” Proc. Steklov Inst. Math. 100, 1–91 (1974).

    MathSciNet  MATH  Google Scholar 

  26. T. Carleman, “Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen,” Ark. Mat. Astron. Fys. 17 (9), 1–30 (1923).

    MATH  Google Scholar 

  27. S. N. Mergelyan, “On completeness of systems of analytic functions,” Uspekhi Mat. Nauk 8 (4 (56)), 3–63 (1953).

    MathSciNet  Google Scholar 

  28. V. I. Smirnov and N. A. Lebedev, Constructive Theory of Functions of a Complex Variable (Nauka, Moscow–Leningrad, 1964) [in Russian].

    MATH  Google Scholar 

  29. M. Keldych, “Sur l’approximation en moyenne par polynômes des fonctions d’une variable complexe,” Rec. Math. [Mat. Sbornik] N. S. 16(58) (1), 1–20 (1945).

    MathSciNet  Google Scholar 

  30. V. L. Goncharov, Theory of Interpolation and Approximation of Functions (GITTL, Moscow, 1954) [in Russian].

    Google Scholar 

  31. V. M. Tikhomirov, Topics in Approximation Theory (Izd. Moskovsk. Univ., Moscow, 1976) [in Russian].

    Google Scholar 

  32. V. N. Temlyakov, Approximation of Periodic Functions (Nova Sci. Publ., Commack, NY, 1993).

    MATH  Google Scholar 

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  1. Alfred Nobel University, 49000, Dnipro, Ukraine

    S. B. Vakarchuk

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Vakarchuk, S.B. Estimates of the Values of \(n\)-Widths of Classes of Analytic Functions in the Weight Spaces \(H_{2,\gamma}(D)\). Math Notes 108, 775–790 (2020). https://doi.org/10.1134/S0001434620110218

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