  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.
