A Functional Decomposition of Finite Bandwidth Reproducing Kernel Hilbert Spaces

TL;DR

This paper studies finite bandwidth reproducing kernel Hilbert spaces, providing conditions for their structure, and explicitly decomposing them, revealing their relation to Hardy spaces and bounded multiplication.

Contribution

It introduces a general matrix recursion approach to characterize finite bandwidth spaces and derives explicit decompositions and boundedness properties for these spaces.

Findings

Point evaluation extends to J+1 points on the circle.

Spaces contain polynomials and have bounded multiplication by z.

Explicit functional decomposition for p>1/2.

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, \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.