On the dual space of a Banach space of entire functions

Abstract

Let \( \mathcal{L}_1 \) denote the subspace of \( L_1(\mathbb{R}) \) consisting of the restrictions to \( \mathbb{R} \) of entire functions of exponential type at most \( \pi \), equipped with the \( L_1(\mathbb{R}) \)-norm. In this paper, we describe the dual space \( \mathcal{L}_1' \), showing that it is isomorphic to the Banach space \( \text{BMO}(\mathbb{Z}) \) of sequences \( x\colon \mathbb{Z} \to \mathbb{C} \) with bounded mean oscillation on \( \mathbb{Z} \). This result is an analogue of Fefferman's classical description of the dual of the Hardy space \( H_1(\mathbb{C}_+) \) of functions analytic in the upper half-plane. A central role in the construction of \( \mathcal{L}_1' \) is played by the discrete Hilbert transform.