Sampling theorem for bandlimited Hardy space functions generated by Regge problem

Abstract

The Whittaker–Shannon–Kotelʼnikov (WSK) sampling theorem provides a reconstruction formula for the Paley–Wiener class of bandlimited functions. It is known that the WSK sampling theorem can be obtained from a sampling theorem associated with a linear, self-adjoint, first order boundary-value problem. In the present paper, a generalization of the WSK sampling theorem is extended to the Hardy space of functions in the upper half-plane, H + 2 . A notion of bandlimitedness in H + 2 is defined and a sampling theorem for this class of bandlimited functions is obtained. Analogous to the Paley–Wiener class of bandlimited functions, which is the image under the Fourier transformation of functions with compact supports, the class of bandlimited functions in the Hardy space is the image of vector-functions with compact supports under an integral transformation, which realizes the incoming spectral representation in scattering theory. The boundary-value problem that generates the sampling points is of the second order and non-self-adjoint (the Regge problem). Its eigenvalues (the sampling points) are known in scattering theory as resonances and eigenfunctions are related to the resonance states. It is shown that the sampling points are complex numbers in the upper half-plane that are uniformly separated and symmetric with respect to the imaginary axis. One of the novelties of the paper is that it sheds light on a connection between three different subjects that do not seem to have much in common, i.e., sampling theorems, H p spaces, and the Lax–Phillips scattering theory.