The Kramer Sampling Theorem Revisited

Abstract

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 \(\mathcal {H}\) through an \(\mathcal {H}\)-valued kernel K defined on an appropriate domain.

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Acknowledgements

The authors wish to thank the referees for their valuable and constructive comments. This work has been supported by the grant MTM2009–08345 from the Spanish Ministerio de Ciencia e Innovación (MICINN).

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Correspondence to A. G. García.

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García, A.G., Hernández-Medina, M.A. & Muñoz-Bouzo, M.J. The Kramer Sampling Theorem Revisited. Acta Appl Math 133, 87–111 (2014). https://doi.org/10.1007/s10440-013-9860-1

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Keywords

  • Sampling formulas
  • Kramer kernels
  • Reproducing kernel Hilbert spaces
  • Lagrange-type interpolation series
  • Zero-removing property
  • Semi-inner products
  • Reproducing kernel Banach spaces
  • Reproducing distributions

Mathematics Subject Classification

  • 46E22
  • 42C15
  • 94A20