Sampling in reproducing kernel Banach spaces (Q358246): Difference between revisions

The focus of this paper is sampling in reproducing-kernel Banach spaces, where sampling is meant to be in the sense of the Whittaker-Shannon-Kotel'nikov sampling theorem which states that if a function \(f\) is band-limited to \( [-\sigma ,\sigma ]\), i.e., it is representable as \[ f(t)= \int_{-\sigma}^{\sigma} e^{-ixt}g(x)\,dx \qquad (t\in\mathbb R)\, , \tag{1.1} \] for some function \(g\in L^2(-\sigma,\sigma)\), then \(f\) can be reconstructed from its samples, \(f(k\pi / \sigma).\) The construction formula is \[ f(t)=\sum_{k=-\infty}^{\infty} f\left( t_k \right) \frac{ \sin{\sigma( t-t_k)}}{\sigma( t-t_k)} \qquad (t\in\mathbb R), \tag{1.2} \] where \(t_k= {k\pi}/{\sigma}.\) The WSK theorem has been generalized in a number of different ways and as a result, it can be looked upon from different points of view as a special case of different settings. For example, it is a special case of Kramer's sampling theorem, sampling in shift-invariant spaces, and sampling in reproducing-kernel Hilbert spaces. Recently, the notion of reproducing-kernel Banach spaces has been developed by several authors. In this paper the authors adopt the theory of reproducing-kernel Banach spaces as introduced by \textit{H. Zhang}, \textit{Y. Xu} and \textit{J. Zhang} [J. Mach. Learn. Res. 10, 2741--2775 (2009; Zbl 1235.68217)] to derive a sampling theorem in the setting of reproducing-kernel Banach spaces. Having established that, they proceed to derive a Kramer-type sampling theorem in reproducing-kernel Banach spaces and a sampling theorem for \(L^p\) shift-invariant spaces.