Let @math be a finite abelian group and @math be a circular convolution operator on @math . The problem under consideration is how to construct minimal @math and @math such that @math is a frame for @math , where @math is the canonical basis of @math . This problem is motivated by the spatiotemporal sampling problem in discrete spatially invariant evolution processes. We will show that the cardinality of @math should be at least equal to the largest geometric multiplicity of eigenvalues of @math , and consider the universal spatiotemporal sampling sets @math for convolution operators whose eigenvalues subject to the same largest geometric multiplicity. We will give an algebraic characterization for such sampling sets and show how this problem is linked with sparse signal processing theory and polynomial interpolation theory.
@cite_35: 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 BJ. 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 BJfor 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.
Our work is closely related to @cite_35 , in which the authors have studied the universal spatial sensor locations for discrete bandlimited space; in some sense, finding universal spatiotemporal sampling sets for convolution operators with eigenvalues subject to the same largest geometric multiplicity in our problem, is analogous to finding universal spatial sampling sets for discrete bandlimited space @math that is subject to the same cardinality of @math in @cite_35 . However, we do not make sparsity assumptions on the signal space. Instead, we seek sub-Nyquist spatial sampling rate, but want to compensate the insufficient spatial sampling rate by oversampling in time.