Random sampling in weighted reproducing kernel subspaces of L^p_ν(R^d)
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In this paper, we mainly study the random sampling and reconstruction for signals in a reproducing kernel subspace of L^p_ν(R^d) without the additional requirement that the kernel function has polynomial decay. The sampling set is independently and randomly drawn from a general probability distribution over R^d, which improves and generalizes the common assumption of uniform distribution on a cube. Based on a frame characterization of reproducing kernel subspaces, we first approximate the reproducing kernel space by a finite dimensional subspace on any bounded domains. Then, we prove that the random sampling stability holds with high probability for all signals in reproducing kernel subspaces whose energy concentrate on a cube when the sampling size is large enough. Finally, a reconstruction algorithm based on the random samples is given for functions in the corresponding finite dimensional subspaces.
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