A Functional Decomposition of Finite Bandwidth Reproducing Kernel Hilbert Spaces

Abstract

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 $w_1,w_2,\dots w_J$ 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 ${\lim }_{n\to \infty }n(1-a_n)=p$. 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.

arXiv comment19 pages with references