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 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.