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Application of Dzyadyk’s Polynomial Kernels in the Constructive Function Theory

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Ukrainian Mathematical Journal Aims and scope

This is a survey of recent results in the constructive theory of functions of complex variable obtained by the author through the application of the theory of Dzyadyk’s kernels combined with the methods and results from modern geometric function theory and the theory of quasiconformal mappings.

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References

  1. J. M. Anderson and W. H. J. Fuchs, “Remarks on “almost best” approximation in the complex plane,” in: Complex Anal., Eds J. Herscch and A. Huber (1988), pp. 17–25.

  2. V. V. Andrievskii, “Direct theorems of approximation theory on quasiconformal arcs,” Math. Izvestiya (USSR), 16, 221–238 (1981).

    Article  Google Scholar 

  3. V. V. Andrievskii, “Approximation characterization of the classes of functions on continua of the complex plane,” Math. Sb., 53, 69–87 (1986).

    Article  Google Scholar 

  4. V. V. Andrievskii, “Approximation of functions by reciprocals of polynomials on a quasismooth arc,” SIAM J. Math. Anal., 44, 2329–2343 (2012).

    Article  MathSciNet  Google Scholar 

  5. V. V. Andrievskii, “Polynomial approximation on touching domains in the complex plane,” Comput. Methods Funct. Theory, 15, 507–527 (2015).

    Article  MathSciNet  Google Scholar 

  6. V. V. Andrievskii, V. I. Belyi, and V. K. Dzjadyk, Conformal Invariants in Constructive Theory of Functions of Complex Variable, World Federation Publ., Atlanta, Georgia (1995).

    MATH  Google Scholar 

  7. V. V. Andrievskii and H.-P. Blatt, Discrepancy of Signed Measures and Polynomial Approximation, Springer, Berlin–New York (2002).

    Book  Google Scholar 

  8. B. Boehm, “Convergence of best rational Chebyshev approximations,” Trans. Amer. Math. Soc., 115, 388–399 (1965).

    Article  MathSciNet  Google Scholar 

  9. V. K. Dzyadyk, “On the theory of approximation of functions on closed sets in the complex plane (with reference to a problem of S. M. Nikol’skii),” Proc. Steklov Inst. Math., 134, 75–130 (1975).

    Google Scholar 

  10. V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).

  11. D. Gaier, “Complex approximation on touching domains,” Complex Variables, 34, 325–342 (1997).

    MathSciNet  MATH  Google Scholar 

  12. R. Grothmann and E. B. Saff, “On the behavior of zeros and poles of best uniform polynomial and rational approximants,”in: A. Cuyt (Ed.), Nonlinear Numerical Methods and Rational Approximation, Reidel, Dordrecht (1988), pp. 57–75.

    Chapter  Google Scholar 

  13. L. I. Kolesnik, “On the approximation of functions continuous on Jordan arcs,” Ukr. Math. Zh., 19, No. 2, 30–37 (1967); English translation: Ukr. Math. J., 19, No. 2, 152–158 (1967).

  14. N. A. Lebedev and N. A. Shirokov, “On uniform approximation on closed sets with finitely many points with nonzero external angles,” Izv. AN Arm. SSR, Ser. Mat., 6, 311–341 (1971).

    MATH  Google Scholar 

  15. D. Leviatan and D. Lubinsky, “Degree of approximation by rational functions with prescribed numerator degree,” Canad. Math. J., 46, 619–633 (1994).

    Article  MathSciNet  Google Scholar 

  16. A. L. Levin and E. B. Saff, “Degree of approximation of real functions by reciprocals of real and complex polynomials,” SIAM J. Math. Anal., 19, 233–245 (1988).

    Article  MathSciNet  Google Scholar 

  17. A. L. Levin and E. B. Saff, “Jackson type theorems in approximation by reciprocals of polynomials,” Rocky Mountain J. Math., 19, 243–249 (1989).

    Article  MathSciNet  Google Scholar 

  18. K. N. Lungu, “Best approximation of |x| by rational functions of the form 1/P n(x),” Sib. Math. J., 15, 1152–1156 (1974).

    MATH  Google Scholar 

  19. J. I. Mamedkhanov, “Problems of the best polynomial approximation in the complex plane,” in: Theory of Functions and Approximation: Proc. Sarat. Winter School [in Russian], Saratov (1983), pp. 149–156.

  20. S. N. Mergelyan, “Uniform approximation of functions of complex variable,” Uspekhi Mat. Nauk, 7, 31–122 (1952).

    MathSciNet  Google Scholar 

  21. D. J. Newman and A. R. Reddy, “Rational approximation to |x|/(1 + x 2m) on (−∞,∞),” J. Approx. Theory, 19, 231–238 (1977).

    Article  Google Scholar 

  22. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Berlin–New York (1992).

    Book  Google Scholar 

  23. E. B. Saff and V. Totik, “Polynomial approximation of piecewise analytic functions,” J. Lond. Math. Soc. (2), 39, 487–498 (1989).

    Article  MathSciNet  Google Scholar 

  24. N. A. Shirokov, “Approximating properties of a continuum,” Investigations on Linear Operators and Function Theory. Pt IX. Zap. Nauchn. Sem. LOMI [in Russian], 92, (1979), pp. 241–252.

    MATH  Google Scholar 

  25. V. I. Smirnov and N. A. Lebedev, Functions of a Complex Variable. Constructive Theory, Massachusetts Institute of Technology, Cambridge (1968).

  26. J. L. Walsh, “On approximation to an analytic function by rational functions of best approximation,” Math. Z., 38, 163–176 (1934).

    Article  MathSciNet  Google Scholar 

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Correspondence to V. Andrievskii.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 2, pp. 151–157, February, 2019.

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Andrievskii, V. Application of Dzyadyk’s Polynomial Kernels in the Constructive Function Theory. Ukr Math J 71, 171–178 (2019). https://doi.org/10.1007/s11253-019-01636-5

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  • DOI: https://doi.org/10.1007/s11253-019-01636-5

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