A Cauchy-Riemann equation for generalized analytic functions

Abstract

We denote by T 2 the torus: z = exp iθ, ω = exp iφ, and we fix a positive irrational number α. A α denotes the space of continuous functions f on T 2 whose Fourier coefficient sequence is supported by the lattice half-plane n + mα > 0. R. Arens and I. Singer introduced and studied the space A α , and it turned out to be an interesting generalization of the disk algebra. Here we construct a differential operator X Σ on a certain 3-manifold Σ 0 such that X Σ characterizes A α in a manner analogous to the characterization of the disk algebra by the Cauchy-Riemann equation in the disk.