Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Factorization of functions
in generalized Nevanlinna classes


Author: Charles Horowitz
Journal: Proc. Amer. Math. Soc. 127 (1999), 745-751
MSC (1991): Primary 30D50
DOI: https://doi.org/10.1090/S0002-9939-99-04581-5
MathSciNet review: 1469410
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For functions in the classical Nevanlinna class analytic projection of $\log |f(e^{i \theta })|$ produces $\log F(z)$ where $F$ is the outer part of $f;$ i.e., this projection factors out the inner part of $f$. We show that if $\log |f(z)|$ is area integrable with respect to certain measures on the disc, then the appropriate analytic projections of $\log |f|$ factor out zeros by dividing $f$ by a natural product which is a disc analogue of the classical Weierstrass product. This result is actually a corollary of a more general theorem of M. Andersson. Our contribution is to give a simple one complex variable proof which accentuates the connection with the Weierstrass product and other canonical objects of complex analysis.


References [Enhancements On Off] (What's this?)

  • 1. M. Andersson, Solution formulas for the $\partial {\overline \partial }$ equation and weighted Nevanlinna classes in the polydisc, Bull. Soc. Math. France 109 (1985), 135-154. MR 87b:32030
  • 2. M. Andersson, Values in the interior of the $L^{2}$-minimal solutions of the $\partial \overline \partial $ equation in the unit ball of $C^{n}$, Publ. Mat. Barcelona 32 (2) (1988), 179-189. MR 90c:32028
  • 3. D. Bekolle, Inégalités à poids pour le projecteur de Bergman dans la boule unité de $C^{n}$, Studia Math. 71 (1981), 305-323. MR 83m:32004
  • 4. E. Beller, Zeros of $A^{p}$ functions and related classes of analytic function, Israel J. Math. 22 (1975), 68-80. MR 52:5973
  • 5. E. Beller, Factorization for non-Nevanlinna classes of analytic functions, Israel J. Math. 27 (1977), 320-330. MR 56:620
  • 6. J. Bruna and J. Ortega-Cerda, On $L^{p}$ solutions of the Laplace equation and zeros of holomorphic functions (to appear).
  • 7. F. Forelli and W. Rudin, Projections on spaces of holomorphic functions in balls, Indiana Univ. J. 24,6 (1974), 593-602. MR 50:10332
  • 8. A. Heilper, The zeros of functions in Nevanlinna's area class, Israel J. Math. 34 (1979), 1-11. MR 82c:30046
  • 9. C. Horowitz, Zero sets and radial zero sets in function spaces, J. Analyse Math. 65 (1995), 145-159. MR 96h:30066
  • 10. A. L. Shields and D. L. Williams, Bounded projections, duality and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 287-302. MR 58:7053

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30D50

Retrieve articles in all journals with MSC (1991): 30D50


Additional Information

Charles Horowitz
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel
Email: horowitz@macs.biu.ac.il

DOI: https://doi.org/10.1090/S0002-9939-99-04581-5
Received by editor(s): June 12, 1997
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society