Advertisement

Interpolation by Bounded Analytic Functions and Related Questions

  • Arthur A. Danielyan
Chapter
Part of the Fields Institute Communications book series (FIC, volume 81)

Abstract

The paper investigates some interpolation questions related to the Khinchine–Ostrowski theorem, Zalcman’s theorem on bounded approximation, and Rubel’s problem on bounded analytic functions.

Keywords

Bounded analytic functions Bounded approximation Fatou’s interpolation theorem Gδ set of measure zero 

Mathematics Subject Classification:

30H05 30H10 

Notes

Acknowledgements

The author wishes to thank the referee for some mathematical and stylistic corrections, and D. Savchuk for preparation of the electronic file of Fig. 1. He also wishes to thank S. Gardiner, V. Totik, and L. Zalcman for valuable discussions related to the topic of this paper.

References

  1. 1.
    Brown, L., Hengartner, W., Gauthier, P.M.: Continuous boundary behaviour for functions defined in the open unit disc. Nagoya Math. J. 57, 49–58 (1975). URL http://projecteuclid.org/euclid.nmj/1118795360
  2. 2.
    Buczolich, Z.: Category of density points of fat Cantor sets. Real Anal. Exchange 29(1), 497–502 (2003/04)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Danielyan, A.A.: On a polynomial approximation problem. J. Approx. Theory 162(4), 717–722 (2010). URL http://dx.doi.org/10.1016/j.jat.2009.09.001
  4. 4.
    Danielyan, A.A.: Rubel’s problem on bounded analytic functions. Ann. Acad. Sci. Fenn. Math. 41(2), 813–816 (2016). URL  http://dx.doi.org/10.5186/aasfm.2016.4151
  5. 5.
    Danielyan, A.A.: Fatou’s interpolation theorem implies the Rudin–Carleson theorem. J. Fourier Anal. Appl. 23(2), 656–659 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hayman, W.K.: Research problems in function theory: new problems, pp. 155–180. London Math. Soc. Lecture Note Ser., No. 12. Cambridge Univ. Press, London (1974)Google Scholar
  7. 7.
    Hoffman, K.: Banach spaces of analytic functions. Prentice-Hall Series in Modern Analysis. Prentice-Hall, Inc., Englewood Cliffs, N. J. (1962)zbMATHGoogle Scholar
  8. 8.
    Kolesnikov, S.V.: On the sets of nonexistence of radial limits of bounded analytic functions. Mat. Sb. 185(4), 91–100 (1994). URL http://dx.doi.org/10.1070/SM1995v081n02ABEH003547
  9. 9.
    Privalov, I.I.: Boundary Properties of Analytic Functions, second edn. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad (1950). German trans.: Priwalow, I. I. Randeigenschaften analytischer Funktionen. Zweite, under Redaktion von A. I. Markuschewitsch überarbeitete und ergänzte Auflage. Hochschulbücher für Mathematik, Bd. 25. VEB Deutscher Verlag der Wissenschaften, Berlin, 1956Google Scholar
  10. 10.
    Zalcman, L.: Polynomial approximation with bounds. J. Approx. Theory 34(4), 379–383 (1982). URL http://dx.doi.org/10.1016/0021-9045(82)90080-6
  11. 11.
    Zygmund, A.: Trigonometric series. 2nd ed. Vol. I. Cambridge University Press, New York (1959)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of South FloridaTampaUSA

Personalised recommendations