## Arkiv för Matematik

### Density of the polynomials in Hardy and Bergman spaces of slit domains

John R. Akeroyd

#### Abstract

It is shown that for any t, 0< t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that $\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\}$ and with the property that the analytic polynomials are dense in the Bergman space $\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma)$ . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in $H^{t}(\mathbb{D}\setminus\Gamma)$ ; improving upon a result in an earlier paper.

#### Article information

Source
Ark. Mat., Volume 49, Number 1 (2011), 1-16.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485907126

Digital Object Identifier
doi:10.1007/s11512-009-0110-8

Mathematical Reviews number (MathSciNet)
MR2784254

Zentralblatt MATH identifier
1223.30017

Rights

#### Citation

Akeroyd, John R. Density of the polynomials in Hardy and Bergman spaces of slit domains. Ark. Mat. 49 (2011), no. 1, 1--16. doi:10.1007/s11512-009-0110-8. https://projecteuclid.org/euclid.afm/1485907126

#### References

• Akeroyd, J., Density of the polynomials in the Hardy space of certain slit domains, Proc. Amer. Math. Soc. 115 (1992), 1013–1021.
• Aleman, A., Richter, S. and Sundberg, C., Nontangential limits in Pt(μ)-spaces and the index of invariant subspaces, Ann. of Math. 169 (2009), 449–490.
• Brennan, J. E., Approximation in the mean by polynomials on non-Carathéodory domains, Ark. Mat. 15 (1977), 117–168.
• Conway, J. B., The Theory of Subnormal Operators, Math. Surveys Monogr. 36, Amer. Math. Soc., Providence, RI, 1991.
• Duren, P. L., Theory ofHpSpaces, Academic Press, New York, 1970.
• Garnett, J. B., Bounded Analytic Functions, Academic Press, Orlando, FL, 1981.
• Garnett, J. B. and Marshall, D. E., Harmonic Measure, Cambridge University Press, New York, 2005.
• Gelbaum, B. R. and Olmstead, J. M. H., Counterexamples in Analysis, Dover, Mineola, NY, 2003.
• Hastings, W. W., A construction of Hilbert spaces of analytic functions, Proc. Amer. Math. Soc. 74 (1979), 295–298.
• Mergelyan, S. N., On the completeness of systems of analytic functions, Uspekhi Mat. Nauk 8 (1953), 3–63 (Russian). English transl.: Amer. Math. Soc. Transl. 19 (1962), 109–166.
• Shields, A. L., Weighted shift operators and analytic function theory, in Topics in Operator Theory, Math. Surveys 13, pp. 49–128, Amer. Math. Soc., Providence, RI, 1974.
• Thomson, J. E., Approximation in the mean by polynomials, Ann. of Math. 133 (1991), 477–507.