On the range of composition operators on spaces of entire functions

TL;DR

This paper characterizes the subspaces generated by composition operators on spaces of entire functions, extending classical results related to bandlimited functions and Paley-Wiener spaces to more general function spaces.

Contribution

It identifies the subspaces of L^2(R) generated by composition operators on entire function spaces and extends results to deBranges-Rovnyak spaces.

Findings

Spaces of bandlimited functions are invariant only under affine composition maps.

Extension of classical theorems to deBranges-Rovnyak spaces.

Abstract

The celebrated Paley-Wiener theorem naturally identifies the spaces of bandlimited functions with subspaces of entire functions of exponential type. Recently, it has been shown that these spaces remain invariant only under composition with affine maps. After some motivation demonstrating the importance of characterization of range spaces of bandlimited functions, in this paper we identify the subspaces of $ L^2 (\mathbb{R}) $ generated by these action. Extension of these theorems where Paley-Wiener spaces are replaced by the deBranges-Rovnyak spaces are given.