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In this paper we deal with the problem of describing the dual space$(B^1_\kappa)^*$ of the Bernstein space $B^1_\kappa$, that is the space ofentire functions of exponential type at most $\kappa>0$ whose restriction tothe real line is Lebesgue integrable. We provide several characterisations,showing that such dual space can be described as a quotient of the space ofentire functions of exponential type $\kappa$ whose restrictions to the realline is Lebesgue integrable. We provide several characterisations, showing thatsuch dual space can be described as a quotient of the space of entire functionsof exponential type $\kappa$ whose restrictions to the real line is in asuitable $BMO$-type space, or as the space of symbols $b$ for which the Hankeloperatorc $H_b$ is bounded on the Paley-Wiener space $B^2_{\kappa/2}$. We alsoprovide a characterisation of $(B^1_\kappa)^*$ as the $BMO$ space w.r.t. theClark measure of the inner function $e^{i\kappa z}$ on the upper half-plane, inanalogy with the known description of the dual of backward-shift invariant$1$-spaces on the torus. Furthermore, we show that the orthogonal projection$P_\kappa\ : L^2(R)\to B^2_\kappa$ induces a bounded operator from$L^\infty(R)$ onto $(B^1_\kappa)^*$. Finally, we show that $B^1_\kappa$ is thedual space of the suitable $VMO$-type space or as the space of symbols $b$ forwhich the Hankel opertor $H_b$ on the Paley-Wiener space $B^2_{k/2}$ iscompact.

Duality, $BMO$ and Hankel operators on Bernstein spaces