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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
Published electronically: December 18, 2009
MathSciNet review: 2587451
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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.

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John Wermer
Affiliation: Department of Mathematics, Brown University, 151 Thayer Street, Providence, Rhode Island 02912

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.

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