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The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 217–223 | Cite as

Equivalent Bergman Spaces with Inequivalent Weights

  • Blake J. BoudreauxEmail author
Article
  • 96 Downloads

Abstract

We give a proof that every space of weighted square-integrable holomorphic functions admits an equivalent weight whose Bergman kernel has zeroes. Here the weights are equivalent in the sense that they determine the same space of holomorphic functions. Additionally, a family of radial weights in \(L^1(\mathbb {C})\) whose associated Bergman kernels that have infinitely many zeroes is exhibited.

Keywords

Admissible weight Bergman space Bergman kernel Radial weight 

Mathematics Subject Classification

Primary 32A25 Secondary 32A36 

References

  1. 1.
    Bommier-Hato, H., Engliš, M., Youssfi, E.: Bergman-type projections in generalized Fock spaces. J. Math. Anal. Appl. 389(2), 1086–1104 (2012).  https://doi.org/10.1016/j.jmaa.2011.12.045 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Conway, J.B.: Functions of One Complex Variable. Graduate Texts in Mathematics, vol. 11, 2nd edn. Springer, New York (1978).  https://doi.org/10.1007/978-1-4612-6313-5 CrossRefGoogle Scholar
  3. 3.
    Duran, Antonio J.: The Stieltjes moments problem for rapidly decreasing functions. Proc. Am. Math. Soc. 107(3), 731–741 (1989).  https://doi.org/10.2307/2048172 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions Related Topics and Applications. Springer Monographs in Mathematics. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-43930-2 CrossRefzbMATHGoogle Scholar
  5. 5.
    Krantz, S.G.: Function Theory of Several Complex Variables. AMS Chelsea Publishing, Providence, RI. Reprint of the 1992 edition. (2001).  https://doi.org/10.1090/chel/340
  6. 6.
    Lu, Q-k: On Kaehler manifolds with constant curvature. Chin. Math. Acta 8, 283–298 (1966)MathSciNetGoogle Scholar
  7. 7.
    Pasternak-Winiarski, Z., Wójcicki, P.M.: Weighted generalization of the Ramadanov theorem and further considerations (2016). ArXiv: 1612.05619
  8. 8.
    Pasternak-Winiarski, Z.: On the dependence of the reproducing kernel on the weight of integration. J. Funct. Anal. 94(1), 110–134 (1990).  https://doi.org/10.1016/0022-1236(90)90030-O MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Pasternak-Winiarski, Z.: On weights which admit the reproducing kernel of Bergman type. Int. J. Math. Math. Sci. 15(1), 1–14 (1992).  https://doi.org/10.1155/S0161171292000012 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Perälä, A.: Vanishing Bergman kernels on the disk. J. Geom. Anal. (2017).  https://doi.org/10.1007/s12220-017-9885-1
  11. 11.
    Michael Range, R.: Holomorphic Functions and Integral Representations in Several Complex Variables. Graduate Texts in Mathematics, vol. 108. Springer, New York (1986).  https://doi.org/10.1007/978-1-4757-1918-5 CrossRefzbMATHGoogle Scholar
  12. 12.
    Shohat, J.A., Tamarkin, J.D.: The Problem of Moments. American Mathematical Society Mathematical Surveys, vol. I. American Mathematical Society, New York (1943)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

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