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Function Spaces for Sampling Expansions

Chapter

Abstract

In this chapter, we consider a variety of Hilbert and Banach spaces that admit sampling expansions \(f(t) = \sum\nolimits_{n=1}^{\infty }f({t}_{n}){S}_{n}(t)\), where {S n } n=1 is a family of functions that depend on the sampling points {t n } n=1 but not on the function f. Those function spaces, that arise in connection with sampling expansions, include reproducing kernel spaces, Sobolev spaces, Wiener amalgam space, shift-invariant spaces, translation-invariant spaces, and spaces modeling signals with finite rate of innovation. Representative sampling theorems are presented for signals in each of these spaces. The chapter also includes recent results on nonlinear sampling of signals with finite rate of innovation, convolution sampling on Banach spaces, and certain foundational issues in sampling expansions.

Keywords

Sobolev Space Reproduce Kernel Hilbert Space Sampling Theorem Finite Rate Wiener Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The second author is partially supported by the National Science Foundation (DMS-1109063).

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Copyright information

© Springer New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Central FloridaOrlandoUSA

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