Sampling in reproducing kernel Banach spaces
type: Item
instance of: scholarly article
title: Sampling in reproducing kernel Banach spaces
zbMATH Open document ID: 1279.42035
author: A. G. Garcia
author: A. Portal
published in: Mediterranean Journal of Mathematics
publication date: 2013-08-16T00:00:00Z
review text: 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.
reviewed by: Ahmed I. Zayed
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 46C50
Mathematics Subject Classification ID: 94A20
zbMATH DE Number: 6199101
zbMATH Keywords: Reproducing kernel Banach spaces
zbMATH Keywords: semi-inner product
zbMATH Keywords: Shift-invariant spaces
zbMATH Keywords: average sampling
MaRDI profile type: Publication
full work available at URL: s00009-012-0234-0
OpenAlex ID: W1981495585
cites work: On stability of sampling-reconstruction models
cites work: Nonuniform sampling and reconstruction in shift-invariant spaces
cites work: Convolution, average sampling, and a Calderon resolution of the identity for shift-invariant spaces
cites work: Theory of Reproducing Kernels
cites work: An introduction to frames and Riesz bases
cites work: On the localization and directionalization of uniform convexity
cites work: Whittaker-Kotelnikov-Shannon sampling theorem and aliasing error
cites work: Representation of linear functionals in a Banach space
cites work: Orthogonal Sampling Formulas: A Unified Approach
cites work: Approximation from shift-invariant spaces by generalized sampling formulas
cites work: Sampling in the functional hilbert space induced by a hilbert space valued kernel
cites work: Q3620490
cites work: Regular multivariate sampling and approximation in \(L^p\) shift-invariant spaces
cites work: Classes of Semi-Inner-Product Spaces
cites work: Wiener’s Lemma: Theme and Variations. An Introduction to Spectral Invariance and Its Applications
cites work: Sampling Expansions in Reproducing Kernel Hilbert and Banach Spaces
cites work: Q4839147
cites work: Q3975831
cites work: Q5332416
cites work: Semi-Inner-Product Spaces
cites work: General sampling theorems for functions in reproducing kernel Hilbert spaces
cites work: Q4340096
cites work: Q4347099
cites work: Q3955832
cites work: Q4319646
cites work: Reproducing kernel Banach spaces for machine learning
cites work: Frames, Riesz bases, and sampling expansions in Banach spaces via semi-inner products
cites work: Q3254471
DOI: 10.1007/S00009-012-0234-0
Recommended article: Sampling Expansions in Reproducing Kernel Hilbert and Banach Spaces
Recommended article: Sampling Theory and Reproducing Kernel Hilbert Spaces
Recommended article: Sampling in Reproducing Kernel Hilbert Space
Recommended article: Q5451878
Recommended article: A converse sampling theorem in reproducing kernel Banach spaces
Recommended article: Sampling in reproducing kernel Banach spaces on Lie groups
Recommended article: Random sampling in reproducing kernel subspaces of \(L^p(\mathbb{R}^n)\)
Recommended article: General sampling theorems for functions in reproducing kernel Hilbert spaces
Recommended article: Random sampling in reproducing kernel spaces with mixed norm
Recommended article: Q4859417
description: scientific article; zbMATH DE number 6199101