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Dzydyk's research on the theory of approximation of functions of a complex variable

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Literature cited

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

    Google Scholar 

  2. S. M. Nikol'skii, “On optimal approximation of functions that satisfy a Lipschitz condition by means of polynomials,” Izv. Akad. Nauk SSSR, Ser. Mat.,10, 295–317 (1946).

    Google Scholar 

  3. V. K. Dzydyk, “On a constructive description of functions that satisfy the condition Lip α(0 < α < 1) on a finite interval of the real axis,” Izv. Akad. Nauk SSSR, Ser. Mat.,20, 623–642 (1956).

    Google Scholar 

  4. A. F. Timan, “Strenthening Jackson's theorem on optimal approximation of continuous functions by polynomials on a finite interval of the real axis,” Dokl. Akad. Nauk SSSR,78, 17–20 (1951).

    Google Scholar 

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

    Google Scholar 

  6. V. K. Dzydyk, “Approximation of continuous functions in closed domains with angles and Nikol'skii's problem,” Izv. Akad. Nauk SSSR, Ser. Mat.,26, No. 6, 797–824 (1962).

    Google Scholar 

  7. V. K. Dzydyk, “Theory of approximation of analytic functions continuous in closed domains and Nikol'skii's problem,” Izv. Akad. Nauk SSSR, Ser. Mat.,27, No. 5, 1135–1164 (1963).

    Google Scholar 

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

    Google Scholar 

  9. V. K. Dzydyk, “Theorems on transformation and approximation of analytic functions,” Dokl. Akad. Nauk SSSR,151, No. 2, 269–272 (1963).

    Google Scholar 

  10. V. K. Dzydyk, “Polynomial approximation of functions in classes Hω on closed domains,” in: Investigations on Modern Problems of the Theory of Functions [in Russian], Izd. Akad. Nauk AzSSR, Baku (1965), pp. 273–283.

    Google Scholar 

  11. V. K. Dzydyk, “Approximation of analytic functions in domains with smooth or piecewisesmooth boundary,” Proceedings of the Third Summer Mathematics School, Inst. Mat. Akad. Nauk UkrSSR, Kiev (1966), pp. 29–83.

    Google Scholar 

  12. V. K. Dzydyk, “Analytic and harmonic transformations and the approximation of harmonic functions,” Ukr. Mat. Zh.,19, No. 5, 33–57 (1967).

    Google Scholar 

  13. N. A. Lebedev and P. M. Tamrazov, “Inverse approximation theorems on closed sets of the complex plane,” Dokl. Akad. Nauk SSSR,179, No. 5, 1046–1049 (1968).

    Google Scholar 

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

    Google Scholar 

  15. V. K. Dzydyk and A. I. Shvai, “Constructive description of functions of Hölder classes in domains with piecewise-smooth boundary and arbitrary positive external corners,” Dokl. Akad. Nauk UkrSSR, Ser. A,11, 1012–1015 (1967).

    Google Scholar 

  16. A. I. Shvai, “Approximation of functions of classes WrHω defined on closed sets with arbitrary (but nonzero and not equal to 2π) angles,” Author's Abstract of Candidate's Dissertation, Mathematics and Physics, Kiev (1967).

  17. R. V. Polyakov, “Approximation of continuous functions defined on bounded closed sets,” Author's Abstract of Candidate's Dissertation, Mathematics and Physics, Kiev (1969).

  18. V. K. Dzydyk and A. I. Shvai, “Approximation of functions of Hölder classes on closed sets with exterior acute angles,” Metricheskie Vopr. Teor. Funkts. Otobrazhenii, No. 2, Naukova Dumka, Kiev (1971), pp. 74–164.

    Google Scholar 

  19. V. K. Dzydyk, “Application of generalized Faber Polynomials to the approximation of Cauchy-type integrals and functions in the class Ar in the domains with smooth or piecewise-smooth boundary,” Ukr. Mat. Zh.,24, No. 1, 3–19 (1972).

    Google Scholar 

  20. N. A. Lebedev and N. A. Shirokov, “Uniform approximation of functions on closed sets with a finite number of angular points with nonzero exterior angles,” Izv. Akad. Nauk ArmSSR,6, No. 4, 311–341 (1971).

    Google Scholar 

  21. I. A. Shevchuk, “Properties of Dzydyk polynomial kernels on the closed interval,” Ukr. Mat. Zh.,41, No. 4, 524–530 (1989).

    Google Scholar 

  22. V. I. Belyi and V. M. Miklyukov, “Certain properties of conformal and quasiconformal mappings and direct theorems of the constructive theory of functions,” Izv. Akad. Nauk SSSR, Ser. Mat.,38, No. 6, 1343–1361 (1974).

    Google Scholar 

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

    Google Scholar 

  24. V. I. Belyi, “Distortion of distances under conformal mappings and direct theorems of the constructive theory of functions,” in: Theory of Approximation of Functions. Proceedings of International Conference on the Theory of Approximation of Functions [in Russian], Nauka, Moscow (1977), pp. 32–37.

    Google Scholar 

  25. V. K. Dzydyk, “Theory of approximation of functions on closed sets of the complex plane (Nikol'skii problem),” Tr. Mat. Inst. Akad. Nauk SSSR,134, 63–114 (1975).

    Google Scholar 

  26. V. I. Belyi, “Conformal mappings and approximation of functions in domains with quasiconformal boundary,” Mat. Sb.,102, No. 3, 331–361 (1977).

    Google Scholar 

  27. V. I. Belyi, Approximation of functions of classes Ar(¯G) in finite domains with quasi-conformal boundary,” Theory of Functions and Mappings tin Russian], Naukova Dumka, Kiev (1979), pp. 37–46.

    Google Scholar 

  28. I. A. Shevchuk, “Classes of functionsH ϕ(l)k,M on setsM of the complex plane,” in: Theory of Approximation of Functions. Proceedings of International Conference on the Theory of Approximation of Functions, Nauka, Moscow (1977), pp. 409–412.

    Google Scholar 

  29. I. A. Shevchuk, “Constructive description of functions continuous on the setM⊂ℂ for the k-th modulus of continuity,” Mat. Zametki,25, No. 2, 225–247 (1979).

    Google Scholar 

  30. V. V. Andrievskii, “Direct theorems of the theory of approximation on quasiconformal arcs,” Izv. Akad. Nauk SSSR SSSR, Ser. Mat.,44, No. 2, 243–261 (1980).

    Google Scholar 

  31. V. V. Andrievskii, “Geometric properties of Dzydyk domains,” Ukr. Mat. Zh.,33, No. 6, 723–727 (1981).

    Google Scholar 

  32. V. V. Andrievskii, “Approximative characterization of classes of functions on continua in the complex plane,” Mat. Sb.,125, No. 1, 70–87 (1984).

    Google Scholar 

  33. V. V. Andrievskii, “Geometric structure of domains and direct theorems of the constructive theory of functions,” Mat. Sb.,126, No. 1, 41–58 (1985).

    Google Scholar 

  34. V. I. Belyi, “The application of conformal invariants to the approximation problems of functions of the complex variable,” Colloquia Math. Soc. Janos Bolyai, Vol. 19, Fourier Analysis and Approximation Theory, Budapest (1976), pp. 113–123.

    Google Scholar 

  35. V. K. Dzydyk, “Approximation method for the approximation of the solutions of linear differential equations by algebraic polynomials,” Izv. Akad. Nauk SSSR, Ser. Mat.,38, No. 4, 937–967 (1974).

    Google Scholar 

  36. V. K. Dzydyk and L. I. Filozof, “Rate of convergence of Padé approximations for certain elementary functions,” Mat. Sb.,107, No. 3, 347–363 (1978).

    Google Scholar 

  37. V. K. Dzydyk, “Asymptotics of diagonal Padé approximations of the functions sin z, cos z, sinh z, cosh z,” Mat. Sb.,108, No. 2, 247–267 (1979).

    Google Scholar 

  38. V. K. Dzydyk, Approximation Methods for the Solution of Differential and Integral Equations [in Russian], Naukova Dumka, Kiev (1988).

    Google Scholar 

  39. V. K. Czydyk, “Generalization of the problem of moments,” Dokl. Akad. Nauk UkrSSR, Ser. A, No. 6, 8–12 (1981).

    Google Scholar 

  40. V. K. Dzydyk, “Generalized problem of moments and Padé approximation,” Ukr. Zh.,35, No. 3, 297–302 (1983).

    Google Scholar 

  41. P. L. Chebyshev, Selected Works [in Russian], N. I. Akhiezer (ed.), Gostekhteoretizdat, Moscow-Leningrad (1946); “Brief survey of Chebyshev's studies,” 178–194.

    Google Scholar 

  42. V. K. Dzydyk and A. P. Golub, Generalized Problem of Moments and Padé Approximation [in Russian], Kiev (1981); Preprint 81∶58 of the institute of Mathematics, Academy of Sciences of the Ukrainian SSR.

  43. M. N. Chyp, Generalized Problem of Moments and Integral Representations of Functions [in Russian], Kiev (1985); Preprint 85∶49 of the Institute of Mathematics, Academy of Sciences of the Ukrainian SSR.

  44. A. F. Leont'ev, “Representation of functions by a sequence of Dirichlet polynomials,” Mat. Sb.,70, No. 1, 122–144 (1966).

    Google Scholar 

  45. A. F. Leont'ev, “Representation of functions by generalized Dirichlet series,” Usp. Mat. Nauk,24, No. 2, 97–164 (1969).

    Google Scholar 

  46. A. F. Leont'ev, “Representation of analytic functions by Dirichlet series,” Mat. Sb.,80, No. 1, 117–150 (1969).

    Google Scholar 

  47. A. F. Leont'ev, “Representation of analytic functions in a closed domain by Dirichlet series,” Izv. Akad. Nauk SSSR, Ser. Mat.,37, No. 3, 577–592 (1973).

    Google Scholar 

  48. V. K. Dzydyk and E. K. Krutigolova, “Representation of analytic functions by Dirichlet series on the boundary of the domain of convergence,” Mat. Zametki,14, No. 6, 769–780 (1973).

    Google Scholar 

  49. V. K. Dzydyk, “Conditions of convergence of Dirichlet series on closed polygons,” Mat. Sb.,95, No. 4, 475–493 (1974).

    Google Scholar 

  50. A. F. Leont'ev, “Representation of analytic functions in the form of a sum of periodic functions,” Mat. Sb.,93, No. 4, 512–528 (1972).

    Google Scholar 

  51. Yu. I. Mel'nik, “Representation of regular functions by Dirichlet series in a closed disk,” Mat. Sb.,94, No. 4, 493–501 (1975).

    Google Scholar 

  52. Yu. I. Mel'nik, “Representation of regular functions by Dirichlet series in closed convex polygons,” Ukr. Mat. Zh.,29, No. 6, 826–830 (1977).

    Google Scholar 

  53. Yu. I. Mel'nik, “Representation of regular functions in the form of a sum of periodic functions,” Mat. Zametki,36, No. 6, 847–856 (1984).

    Google Scholar 

  54. A. M. Sedletskii, “Bases formed by exponential functions in spaces EP on convex polygons,” Izv. Akad. Nauk SSSR, Ser. Mat.,42, No. 5, 1101–1119 (1978).

    Google Scholar 

  55. A. M. Sedletskii, “Expansions of functions in Dirichlet series on closed convex polygons,” Sib. Mat. Zh.,19, No. 4, 878–887 (1978).

    Google Scholar 

  56. A. M. Sedletskii, “Expansion of analytic functions into a sum of periodic functions,” Izv. Akad. Nauk SSSR, Ser. Mat.,48, No. 4, 833–853 (1984).

    Google Scholar 

  57. V. K. Dzjadyk (Dzydyk), “On a problem of Chebyshev and Markov,” Anal. Math.,3, No. 3, 171–175 (1977).

    Google Scholar 

  58. V. K. Dzydyk, “Iterative-approximation method of approximation the solution of Cauchy's problem for ordinary differential equations by means of polynomials,” Zh. Vychislit. Mat. Mat. Fiz.,26, No. 3, 357–372 (1986).

    Google Scholar 

  59. N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  60. V. K. Dzydyk, “Theory of special functions and their approximation,” Mat. Sb.,131, No. 4, 438–464 (1986).

    Google Scholar 

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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 4, pp. 441–454, April, 1989.

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Belyi, V.I., Golub, A.P. & Shevchuk, I.A. Dzydyk's research on the theory of approximation of functions of a complex variable. Ukr Math J 41, 384–395 (1989). https://doi.org/10.1007/BF01060614

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