  In this work, we consider "finite bandwidth" reproducing kernel Hilbert
spaces which have orthonormal bases of the form $f_n(z)=z^n \prod_{j=1}^J
\left( 1 - a_{n}w_j z \right)$, where $w_1 ,w_2, \ldots w_J $ are distinct
points on the circle $\mathbb{T}$ and $\{ a_n \}$ is a sequence of complex
numbers with limit $1$. We provide general conditions based on a matrix
recursion that guarantee such spaces contain a functional multiple of the Hardy
space. Then we apply this general method to obtain strong results for finite
bandwidth spaces when $\lim_{n\rightarrow \infty} n (1-a_n)=p$. In particular,
we show that point evaluation can be extended boundedly to precisely $J$
additional points on $\mathbb{T}$ and we obtain an explicit functional
decomposition of these spaces for $p>1/2$ in analogy with a previous result in
the tridiagonal case due to Adams and McGuire. We also prove that
multiplication by $z$ is a bounded operator on these spaces and that they
contain the polynomials.
