On the dual space of a Banach space of entire functions - Mykytyuk, Ya.; Sushchyk, N.; Lukivska, D. (Matematychni Studii, 2024)
Let \( L_1 \) denote the subspace of \( L_1(R) \) consisting of the restrictions to \( R \) of entire functions of exponential type at most \( π \), equipped with the \( L_1(R) \)-norm. In this paper, we describe the dual space \( L_1’ \), showing that it is isomorphic to the Banach space \( BMO(Z) \) of sequences \( xZ → C \) with bounded mean oscillation on \( Z \). This result is an analogue of Fefferman’s classical description of the dual of the Hardy space \( H_1(C_+) \) of functions analytic in the upper half-plane. A central role in the construction of \( L_1’ \) is played by the discrete Hilbert transform.