# An algebra of polyanalytic functions

## TL;DR

This paper introduces a new algebraic framework for polyanalytic functions on compact sets, extending classical uniform algebras of analytic functions with a novel multiplication structure.

## Contribution

It develops a Banach space structure for polyanalytic functions and defines a new multiplication, creating a new class of algebras for these functions.

## Key findings

- Defined a multiplication on polyanalytic function spaces
- Established Banach space properties for these spaces
- Analyzed the algebraic structure resulting from the new multiplication

## Abstract

The most important uniform algebra is the family of continuous functions on a compact subset $K$ of the complex plane $\mathbb{C}$ which are analytic on the interior int$(K)$ For compact sets $K$ which are regular (i.e. $K =$int$(K)$ and for polyanalytic functions, we introduce analogous spaces, which are Banach spaces with respect to the sup-norm, but are not closed with respect to the usual pointwise multiplication. We shall introduce a multiplication on these spaces and investigate the resulting algebras.