Factorial and Noetherian subrings of power series rings

Let $F$ be a field. We show that certain subrings contained between the polynomial ring $F[X] = F[X_1, \cdots, X_n]$ and the power series ring $F[X][[Y]] = F[X_1, \cdots, X_n][[ Y]]$ 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 the $p$-adic analytic variation of zeta functions over finite fields. Future research into this area may study more complicated subrings in which $Y = (Y_1, \cdots, Y_m)$ has more than one variable, and for which there are multiple degree functions, $\lambda_1, \cdots, \lambda_m$. Another direction of study would be to generalize these results to $k$-affinoid algebras.