We study the problem of interpolating all values of a discrete signal f of length N when d < N values are known, especially in the case when the Fourier transform of the signal is zero outside some prescribed index set J ; these comprise the (generalized) bandlimited spaces B J . The sampling pattern for f is specified by an index set I, and is said to be a universal sampling set if samples in the locations I can be used to interpolate signals from B J for any J . When N is a prime power we give several characterizations of universal sampling sets, some structure theorems for such sets, an algorithm for their construction, and a formula that counts them. There are also natural applications to additive uncertainty principles.

We study the problem of finding unitary submatrices of the N × N discrete Fourier transform matrix, in the context of interpolating a discrete bandlimited signal using an orthogonal basis. This problem is related to a diverse set of questions on idempotents on Z N and tiling Z N . In this work, we establish a graph-theoretic approach and connections to the problem of finding maximum cliques. We identify the key properties of these graphs that make the interpolation problem tractable when N is a prime power, and we identify the challenges in generalizing to arbitrary N . Finally, we investigate some connections between graph properties and the spectral-tile direction of the Fuglede conjecture.

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