Sampling in Shift-Invariant Spaces Generated by Hilbert Space-Valued Functions

In this paper, we investigate the problem of sampling and reconstruction in principal shift-invariant spaces generated by Hilbert space-valued functions. Given any signal (Formula presented.) and data point (Formula presented.), the sample (Formula presented.) is stored along a sequence of directions (Formula presented.). Specifically, the inner products (Formula presented.) are stored. First, we define what we mean by a stable set of sampling and provide equivalent conditions for proving that a given set is a stable set of sampling. We then present a reconstruction formula for (Formula presented.) from its integer samples (Formula presented.). Finally, we address the cases of perturbed and irregular sampling, examining their impact on the reconstruction process.