Hilbert spaces of distributions having an orthogonal basis of exponentials

We characterize the Hilbert spaces H whose elements are distributions supported on the interval 0, 1] and which have the property that the system of exponentials {e^2πinx}_n∈Z forms a complete orthogonal system for H, generalizing in this way the classical situation where H=L²(0, 1]) and the system is actually orthonormal. This characterization is extended to the more general setting of spectral pairs and is used to obtain sampling results in various related spaces of functions, that generalize the classical Shannon sampling theorem.