Skip to main content
SpringerLink
Log in

The Structure of Certain Spaces of Analytic Functions

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

Let μ be a measure of compact support in the complex plane. In 1991 J. E. Thomson described completely the structure of H p(μ), the closed subspace of L p(μ) spanned by the polynomials, when 1 ≤ p < ∞. Here we discuss the extent to which Rp(μ), the closed subspace of L p(μ) spanned by the rational functions having no poles on the support of μ, admits a similar description.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. E. Brennan, Invariant subspaces and rational approximation, J. Funct. Anal. 7 (1971), 285–310.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. E. Brennan, Thomson’s theorem on mean-square polynomial approximation, Algebra i analiz 17 no.2 (2005), 1–32; English transl.: St. Petersburg Math. J. 17 no.2 (2006), 217–238.

    MathSciNet  Google Scholar 

  3. J. Chaumat, Adhérence faible étoile d’algèbres de fractions rationelles, Ann. Inst. Fourier 24 (1974), 93–120.

    Article  MathSciNet  Google Scholar 

  4. J. Conway, The Theory of Subnormal Operators, Math. Surveys Monogr., vol. 36, Amer. Math. Soc., Providence, RI, 1991.

    MATH  Google Scholar 

  5. J. Conway and N. Elias, Analytic bounded point evaluations for spaces of rational functions, J. Funct. Anal. 117 (1993), 1–24.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. M. Davie, Analytic capacity and approximation problems, Trans. Amer. Math. Soc. 171 (1972), 409–444.

    Article  MathSciNet  MATH  Google Scholar 

  7. C. Fernstrom, Bounded point evaluations and approximation in Lp by analytic functions, Lecture Notes in Math., vol. 512, pp. 65–68, Springer-Verlag, Berlin, 1976.

    Google Scholar 

  8. T. W. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, NJ, 1969.

    MATH  Google Scholar 

  9. M. S. Mel’nikov, On the Gleason parts of the algebra R(X), Mat. Sb. 101 no.2 (1976), 293–300; English transl.: Math. USSR-Sb 30 (1976), 261–268.

    Google Scholar 

  10. J. E. Thomson, Approximation in the mean by analytic functions, Ann. of Math. (2) 133 (1991), 477–507.

    Article  MathSciNet  MATH  Google Scholar 

  11. X. Tolsa, L2-boundedness of the Cauchy integral operator for continuous measures, Duke Math. J. 98 (1999), 269–304.

    Article  MathSciNet  MATH  Google Scholar 

  12. X. Tolsa, Painlevé’s problem and the semiadditivity of analytic capacity, Acta Math. 190 (2003), 105–149.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. G. Vitushkin, Analytic capacity of sets and problems in approximation theory, Uspekhi Mat. Nauk 22 (1967), 141–199; English transl.: Russian Math. Surveys 22 (1967), 139–200.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, Lexington, KY, 40506, USA

    James E. Brennan

Authors
  1. James E. Brennan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James E. Brennan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brennan, J.E. The Structure of Certain Spaces of Analytic Functions. Comput. Methods Funct. Theory 8, 625–640 (2008). https://doi.org/10.1007/BF03321709

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03321709

Keywords

2000 MSC

Search

Navigation