The Banach—Rudin—Carleson Interpolation Theorems and the Norms of Embedding Operators for Some Classes of Analytical Functions

  • S. A. Vinogradov
Part of the Seminars in Mathematics book series (SM)


In the present articlet† we investigate the behavior of the Maclaurin coefficients of analytic functions f continuous‡ in the closed unit disk Open image in new window as a function of the recurrence of the set of singularities Tf of those functions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. 1.
    Banach, S.,Über einige Eigenschaften der lacunaren trigonometrischen Reihen, Studia Math., 2:207–220 (1930).Google Scholar
  2. 2.
    Carleman, T., Über die Fourierkoeffizienten einer stetigen Funktion, Acta Math., 41: 377–384 (1918).CrossRefGoogle Scholar
  3. 3.
    Paley, R. E. A. C., A note on power series, J. London Math. Soc., 7: 122–130 (1932).CrossRefGoogle Scholar
  4. 4.
    Stechkin, S. B., An extremal problem for polynomials, Izv. Akad. Nauk SSSR, Ser. Matem., 20 (6): 765–774 (1956).Google Scholar
  5. 5.
    Khavin, V. P., On the norms of certain operators in polynomial space, Vestnik Leningrad Univ. (LGU), Ser. Matem. Mekh. Astron., 4 (19): 47–59 (1959).Google Scholar
  6. 6.
    Hoffman, K. Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs (N. J. ) (1962).Google Scholar
  7. 7.
    Privalov, I. I., Boundary Properties of Analytic Functions, Moscow-Leningrad (1950).Google Scholar
  8. 8.
    Tumarkin, G. Ts., and Khavinson, S. Ya., Classes of analytic functions in multiply connected domains, in: Studies in Current Problems of the Theory of Functions of a Complex Variable, Fizmatgiz, Moscow-Leningrad (1960), pp. 45–77.Google Scholar
  9. 9.
    Khavinson, S. Ya., Extremal problems for certain classes of analytic functions in finitely connected domains, Matem. Sborn., 36(78)(3): 445–478 (1955).Google Scholar
  10. 10.
    Rudin, W., Trigonometric series with caps, J. Math. Mech., 9 (2): 203–228 (1960).Google Scholar
  11. 11.
    Zygmund, A., Trigonometric Series [Russian translation], Vol. 1, Izd. Mir, Moscow (1965) [English translation: Cambridge Univ. Press (1959)].Google Scholar
  12. 12.
    Kaczmarz, S. and Steinhaus, H., Theory of Orthogonal Series [Russian translation], Fizmatigiz, Moscow (1958).Google Scholar
  13. 13.
    Fikhtengol’ts, G. M., Course in Differential and Integral Calculus, Vol. 2, Fizmatgiz, Moscow (1962).Google Scholar
  14. 14.
    Kantorovich, L. V., and Akilov, G. P., Functional Analysis in Normed Spaces, Fizmatgiz, Moscow (1959).Google Scholar
  15. 15.
    Dunford, N., and Schwartz, J. T. Linear Operators, Vol. 1: General Theory [Russian transla-tion], Izd. In. Lit., Moscow (1962) [English edition: Interscience, New York (1958)].Google Scholar
  16. 16.
    Walsh, J. L., Interpolation and Approximation by Rational Functions in the Complex Domain (AMS Colloquium Publ., Vol. 20 ), Amer. Math. Soc., Providence (R. I. ) (1965).Google Scholar
  17. 17.
    Makhmudov, A. S., On the Fourier and Taylor coefficients of continuous functions, in: Some Problems of Functional Analysis and Its Applications, Baku (1965), pp. 103–128.Google Scholar
  18. 18.
    Makhmudov, A. S., On the Fourier and Taylor coefficients of continuous functions, Izv. Akad. Nauk Azerb. SSR, Ser. Fiz.-Matem. Tekh., Nos. 2 and 4 (1964).Google Scholar
  19. 19.
    Goluzin, G. M., Geometric Theory of Functions of a Complex Variable, Moscow-Leningrad (1952).Google Scholar
  20. 20.
    Vinogradov, S. A., The Paley singularities and Rudin-Carleson interpolation theorems for certain classes of analytic functions, Dokl. Akad. Nauk SSSR, 178 (3): 511–514 (1968).Google Scholar
  21. 21.
    Akilov, G. P., and Vershik, A. M., On the mutually continuous extension of linear operators, Vestnik Leningrad. Univ. (LGU), Ser. Matem. Mekh. Astron., 2 (7) (1958).Google Scholar

Copyright information

© Springer Science+Business Media New York 1972

Authors and Affiliations

  • S. A. Vinogradov

There are no affiliations available

Personalised recommendations