Duality theorems for polyanalytic functions

The goal of this paper is to give a characterization of the dual of the space of polyanalytic functions on the complement of a compact set K and vanishing at infinity. The class of polyanalytic functions generalizes holomorphic functions and serves as a middle ground between holomorphic functions in one complex variable and those in two complex variables. The duality result is also expressed in topological terms through a new class of infinite-order differential operators, which includes well-known families of operators like the Laplace and Helmholtz operators. Since the notion of polyanalytic function generalizes that of holomorphic function, the duality theorems established in this paper can be considered a non-trivial generalization of the K & ouml;the-Grothendieck theorem.