Application of a Riesz-type formula to weighted Bergman spaces

Author:
Ali Abkar

Journal:
Proc. Amer. Math. Soc. **131** (2003), 155-164

MSC (2000):
Primary 31A30; Secondary 30E10, 30H05, 46E10

Published electronically:
May 13, 2002

MathSciNet review:
1929035

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the unit disk in the complex plane. We consider a class of superbiharmonic weight functions whose growth are subject to the condition for some constant . We first establish a Reisz-type representation formula for , and then use this formula to prove that the polynomials are dense in the weighted Bergman space with weight .

**1.**Ali Abkar and Håkan Hedenmalm,*A Riesz representation formula for super-biharmonic functions*, Ann. Acad. Sci. Fenn. Math.**26**(2001), no. 2, 305–324. MR**1833243****2.**A. Aleman, S. Richter, and C. Sundberg,*Beurling’s theorem for the Bergman space*, Acta Math.**177**(1996), no. 2, 275–310. MR**1440934**, 10.1007/BF02392623**3.**Peter L. Duren,*Theory of 𝐻^{𝑝} spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655****4.**P. R. Garabedian,*Partial differential equations*, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR**0162045****5.**V. P. Havin and N. K. Nikolski (eds.),*Linear and complex analysis. Problem book 3. Part II*, Lecture Notes in Mathematics, vol. 1574, Springer-Verlag, Berlin, 1994. MR**1334346****6.**Per Jan Håkan Hedenmalm,*A computation of Green functions for the weighted biharmonic operators Δ\vert𝑧\vert^{-2𝛼}Δ, with 𝛼>-1*, Duke Math. J.**75**(1994), no. 1, 51–78. MR**1284815**, 10.1215/S0012-7094-94-07502-9**7.**Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu,*Theory of Bergman spaces*, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. MR**1758653****8.**Kenneth Hoffman,*Banach spaces of analytic functions*, Dover Publications, Inc., New York, 1988. Reprint of the 1962 original. MR**1102893****9.**S. N. Mergelyan,*On completeness of systems of analytic functions*, Amer. Math. Soc. Translations**19**(1962), 109-166.**10.**Walter Rudin,*Real and complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR**924157**

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Additional Information

**Ali Abkar**

Affiliation:
Department of Mathematics, Imam Khomeini International University, P.O. Box 288, Qazvin 34194, Iran

Address at time of publication:
Department of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-1795, Tehran, Iran

Email:
abkar@ipm.ir

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06491-2

Received by editor(s):
August 16, 2001

Published electronically:
May 13, 2002

Additional Notes:
This research was supported in part by a grant from the Institute for Theoretical Physics and Mathematics (IPM), Tehran, Iran.

Communicated by:
Juha M. Heinonen

Article copyright:
© Copyright 2002
American Mathematical Society