A functional decomposition of finite bandwidth reproducing kernel Hilbert spaces

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 circle $\mathbb{T}$ and $\{ a_n \}$ is a sequence complex numbers with limit $1$. We provide general conditions based matrix recursion that guarantee such contain functional multiple Hardy space. Then apply method to obtain strong results for when $\lim_{n\rightarrow \infty} n (1-a_n)=p$. particular, show point evaluation can be extended boundedly precisely $J$ additional an explicit decomposition these $p>1/2$ in analogy previous result tridiagonal case due Adams McGuire. also prove multiplication by $z$ bounded operator they polynomials.