Sampling theory in abstract reproducing Kernel Hilbert space

Abstract

Let H be a separable Hilbert space and k(t) a H-valued function on a subset Ω of the real line $\mathbb{R}$ such that {k(t) |t ∈ Ω}$ is total in H. Then {f_x(t):={x, k(t)_H|x ∈ H} becomes a reproducing kernel Hilbert space (RKHS) in a natural way. Here, we develop a sampling formula for functions in this RKHS, which generalizes the well-known celebrated Whittaker-Shannon-Kotel``nikov (WSK) sampling formula in the Paley-Wiener space of band-limited signals. To be more precise, we develop a multi-channel sampling formula in which every channel is given arbitrary chosen sampling points.