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Constructive characterization of classes of continuous functions on a set\(\mathfrak{M} \subset C\) for the K-th modulus of continuity

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Literature cited

  1. 1.

    V. K. Dzyadyk, “On the theory of approximation of functions on closed sets of the complex plane (apropos of a problem of S. M. Nikol'skii),” Tr. Mat. Inst. Akad. Nauk SSSR,134, 63–114 (1975).

  2. 2.

    V. K. Dzyadyk, “Application of methods of the theory of quasiconformal mappings to obtain direct theorems of approximation of functions,” Dokl. Akad. Nauk SSSR,225, No. 3, 491–494 (1975).

  3. 3.

    N. K. Bari and S. B. Stechkin, “Best approximations and differential properties of two conjugate functions,” Tr. Mosk. Mat. Obshch.,5, 483–522 (1956).

  4. 4.

    J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Am. Math. Soc., Providence (1966).

  5. 5.

    S. Ya. Al'per, “On uniform approximation of functions of a complex variable in closed domains,” Izv. Akad. Nuak SSSR, Ser. Mat.,19, No. 6, 423–444 (1955).

  6. 6.

    V. V. Ivanov, “Certain properties of singular integrals,” Izv. Vyssh. Uchebn. Zaved.,4, 89–95 (1958).

  7. 7.

    S. M. Nikol'skii, “Approximation of functional of several variables of polynomials,” in: Proceedings of the Third All-union Mathematical Symposium [in Russian], Vol. 3, Izd. Akad. Nauk SSSR, Moscow (1958), pp. 226–231.

  8. 8.

    V. K. Dzyadyk, “On a problem of S. M. Nikol'skii in a complex domain,” Izv. Akad. Nauk SSSR, Ser. Mat.,23, No. 5, 697–736 (1959).

  9. 9.

    V. K. Dzyakyk, “On the problem of approximation of continuous functions in closed domains with corners and on a problem of S. M. Nikol'skii (the first communication),” Izv. Akad. Nauk SSSR, Ser. Mat.,26, No. 6, 796–824 (1962).

  10. 10.

    V. K. Dzyadyk, “Inverse theorems of approximation of functions in complex domains,” Ukr. Mat. Zh.,15, No. 4, 365–375 (1963).

  11. 11.

    V. K. Dzyadyk, “On the theory of approximation of analytic functions, continuous in closed domains, and on a problem of S. M. Nikol'skii (second communication),” Izv. Akad. Nauk SSSR, Ser. Mat.,27 No. 5, 1135–1164 (1963).

  12. 12.

    R. V. Polyakov, “Approximation of continuous (analytic inside) functions in domains with smooth boundary,” Ukr. Mat. Zh.,24, No. 1, 57–68 (1972).

  13. 13.

    I. A. Shevchuk, “On the constructive characterization of functions of the classes on closed sets with piecewise smooth boundary,” Ukr. Mat. Zh.,25, No. 1, 81–90 (1973).

  14. 14.

    N. N. Borob'ev and R. V. Polyakov, “On the constructive characterization of continuous functions, given on smooth arcs,” Ukr. Mat. Zh.,20, No. 6, 750–758 (1968).

  15. 15.

    P. M. Tamrazov, Smoothness and Polynomial Approximations [in Russian], Naukova Dumka, Kiev (1975).

  16. 16.

    E. M. Dyn'kin, “On uniform polynomial approximation in a complex domain,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,47, 164–165 (1974).

  17. 17.

    H. Whitney, “On functions with bounded n-th differences,” J. Math. Pures Appl.36, No. 9, 67–95 (1957).

  18. 18.

    S. B. Stechkin, “On the order of best approximations of continuous functions,” Izv. Akad. Nauk SSSR, Ser. Mat.,15, 219–242 (1951).

  19. 19.

    V. I. Belyi, “On approximation of quasismooth functions of a complex variable,” Izv. Akad. Nauk Arm. SSR, Ser. Mat.,4, No. 5, 364–393 (1969).

  20. 20.

    E. M. Dyn'kin, “On the general problem of approximation by polynomials in Jordan domains,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,65, 189–192 (1976).

  21. 21.

    I. A. Shevchuk, “On the classes of functions\(H_{k, \mathfrak{M}}^{q (t)} \) on sets\(\mathfrak{M}\) of the complex plane,” in: Proceedings of the International Conference on the Theory of Approximation of Functions at Kaluga [in Russian], Nauka, Moscow (1977), pp. 409–412.

  22. 22.

    A. F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon (1973).

  23. 23.

    V. K. Dzyadyk and I. A. Shevchuk, “On the limiting values of Cauchy-type integrals for functions of Zygmund classes,” Ukr. Mat. Zh.,24, No. 5, 601–617 (1972).

  24. 24.

    N. A. Lebedev and P. M. Tamrazov, “Inverse theorems of approximation on regular compacta of the complex plane,” Izv. Akad. Nauk SSSR, Ser. Mat.,34, No. 6, 1340–1390 (1970).

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Translated from Matematicheskie Zametki, Vol. 25, No. 2, pp. 225–247, February, 1979.

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Shevchuk, I.A. Constructive characterization of classes of continuous functions on a set\(\mathfrak{M} \subset C\) for the K-th modulus of continuity. Mathematical Notes of the Academy of Sciences of the USSR 25, 117–129 (1979). https://doi.org/10.1007/BF01142722

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Keywords

  • Continuous Function
  • Constructive Characterization