Sampling theory in abstract reproducing kernel Hilbert space

Abstract

Let H be a separable Hilbert space and k(t) an H-valued function on a subset Ω of the real line R such that {k(t) t ε Ω} is total in H. Then {fx:= (x, k(t))H x ε 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-Kotelnikov sampling formula in the Paley-Wiener space of band-limited signals. To be more precise, we develop a multi-channel sampling formula in which each channel is given a rather arbitrary sampling rate. We also discuss stability and oversampling. ？ 2007 Sampling Publishing.