The Kramer Sampling Theorem Revisited

• Published in: Acta applicandae mathematicae Vol. 133; no. 1; pp. 87 - 111
• Main Authors: García, A. G., Hernández-Medina, M. A., Muñoz-Bouzo, M. J.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-10-2014; Springer Nature B.V
• Subjects: Algebra; Applications of Mathematics; Applied mathematics; Banach spaces; Calculus of Variations and Optimal Control; Optimization; Computational Mathematics and Numerical Analysis; Differential equations; Fourier transforms; Functions (mathematics); Hilbert space; Kernels; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Partial Differential Equations; Probability Theory and Stochastic Processes; Sampling; Studies; Texts; Theorems; Reproducing kernel Banach spaces; Zero-removing property; 46E22; Lagrange-type interpolation series; Reproducing kernel Hilbert spaces; Semi-inner products; Sampling formulas; Reproducing distributions; 42C15; Kramer kernels; 94A20
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-013-9860-1

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. Besides, it has been the cornerstone for a significant mathematical literature on the topic of sampling theorems associated with differential and difference problems. In this work we provide, in an unified way, new and old generalizations of this result corresponding to various different settings; all these generalizations are illustrated with examples. All the different situations along the paper share a basic approach: the functions to be sampled are obtaining by duality in a separable Hilbert space
through an
-valued kernel
K
defined on an appropriate domain.