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 J j=1 (1 − a n w j z), where w 1 , w 2 , . . . w J are distinct points on the circle 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→∞ n(1 − a n ) = p. In particular, we show that point evaluation can be extended boundedly to precisely J additional points on 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.