A Cauchy-Riemann equation for generalized analytic functions

Wermer, John

doi:10.1090/S0002-9939-09-10228-9

A Cauchy-Riemann equation for generalized analytic functions
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Proc. Amer. Math. Soc. 138 (2010), 1667-1672 Request permission

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

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