  Let $F$ be a field. We show that certain subrings contained between the
polynomial ring $F[X] = F[X_1, ..., X_n]$ and the power series ring $F[X][[Y]]
= F[X_1, ..., X_n][[Y]]$ have Weierstrass Factorization, which allows us to
deduce both unique factorization and the Noetherian property. These
intermediate subrings are obtained from elements of $F[X][[Y]]$ by bounding
their total $X$-degree above by a positive real-valued monotonic up function
$\lambda$ on their $Y$-degree. These rings arise naturally in studying $p$-adic
analytic variation of zeta functions over finite fields. Future research into
this area may study more complicated subrings in which $Y = (Y_1, >..., Y_m)$
has more than one variable, and for which there are multiple degree functions,
$\lambda_1, ..., \lambda_m$. Another direction of study would be to generalize
these results to $k$-affinoid algebras.
