We investigate shifts of prolate spheroidal wave functions and more particularly the shift-invariant spaces generated by the shifts of the first N of these functions. The Markov property satisfied by the prolates is used to show that at the Nyquist rate, such collections form a Riesz basis for the associated Paley-Wiener space, although explicit Riesz bounds cannot be derived. The fact that the prolate functions are eigenfunctions of the finite Fourier transform and a quadrature estimate for integrals of complex exponentials is used to provide Riesz bounds when the sampling rate is much lower than Nyquist. In this case, the Riesz basis will typically not span all of the Paley-Wiener space.
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