Polyanalytic Functions on Banach Manifolds

Abstract

We generalize to infinite dimensions the concept of polyanalytic functions.These are our main results: (1) A characterization based upon restriction, which generalizes a known characterization of holomorphic functions on Banach spaces. (2) A property of special local uniform limits which yields an approximation result. (3) We introduce meta-analytic functions on Hilbert manifolds together with a characterization compatible with (1). We also point out several corollaries to our characterizations.

Polyanalytic reproducing kernels in $$\mathbb {C}^n$$

“…Another notion of polyanalyticity was considered recently by Daghighi [5]. Namley, for a positive integer q, a function f defined on an open set Ω in ℂ n is called analytic of absolute order q (or just q-analytic) if it is of class C q that satisfies the generalized Cauchy-Riemann equation…”