A Cauchy-Riemann equation for generalized analytic functions

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 int...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2010-05, Vol.138 (5), p.1667-1672
Main Author: WERMER, JOHN
Format: Article
Language:English
Subjects:
Algebra
Analytic functions
Cauchy Riemann equations
Continuous functions
Coordinate systems
Differential operators
Exact sciences and technology
Functional analysis
Functions of a complex variable
General mathematics
General, history and biography
Integers
Mathematical analysis
Mathematical functions
Mathematical theorems
Mathematics
Real functions
Sciences and techniques of general use
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Summary: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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-09-10228-9