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)

 
 

 

A Cauchy-Riemann equation for generalized analytic functions


Author: John Wermer
Journal: Proc. Amer. Math. Soc. 138 (2010), 1667-1672
MSC (2000): Primary 32-XX
DOI: https://doi.org/10.1090/S0002-9939-09-10228-9
Published electronically: December 18, 2009
MathSciNet review: 2587451
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We denote by $ T^{2}$ the torus: $ z = \exp i\theta, w = \exp i\phi$, and we fix a positive irrational number $ \alpha$. $ A_{\alpha}$ denotes the space of continuous functions $ f$ on $ T^{2}$ whose Fourier coefficient sequence is supported by the lattice half-plane $ n + m\alpha \geq 0$. R. Arens and I. Singer introduced and studied the space $ A_{\alpha}$, and it turned out to be an interesting generalization of the disk algebra. Here we construct a differential operator $ X_{\Sigma}$ on a certain 3-manifold $ \Sigma_{0}$ such that $ X_{\Sigma}$ characterizes $ A_{\alpha}$ in a manner analogous to the characterization of the disk algebra by the Cauchy-Riemann equation in the disk.


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

  • 1. R. Arens and I. Singer, Generalized analytic functions, Trans. Amer. Math. Soc. 81 (1956), 379-393. MR 0078657 (17:1226e)
  • 2. T. W. Gamelin, Uniform Algebras, Prentice Hall, Inc., 1969. MR 0410387 (53:14137)
  • 3. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 2nd edition, Oxford, 1945. MR 0067125 (16:673c)
  • 4. H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math. 99 (1958), 165-202. MR 0097688 (20:4155)
  • 5. K. Hoffman and I. M. Singer, Maximal subalgebras of $ C(\Gamma)$, Amer. Jour. of Math. 79 (1957), 295-305. MR 0085478 (19:46e)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32-XX

Retrieve articles in all journals with MSC (2000): 32-XX


Additional Information

John Wermer
Affiliation: Department of Mathematics, Brown University, 151 Thayer Street, Providence, Rhode Island 02912
Email: wermer@math.brown.edu

DOI: https://doi.org/10.1090/S0002-9939-09-10228-9
Received by editor(s): May 8, 2009
Published electronically: December 18, 2009
Communicated by: Franc Forstneric
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society