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Sampling analysis in the complex reproducing kernel Hilbert space1

Published online by Cambridge University Press:  21 November 2014

BING-ZHAO LI  and
QING-HUA JI
BING-ZHAO LI*
Affiliation:
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China, 100081 Beijing Key Laboratory of Fractional Signals and Systems, Beijing, China, 100081
QING-HUA JI
Affiliation:
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China, 100081
*
*Corresponding Email: [email protected]
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Abstract

We consider and analyse sampling theories in the reproducing kernel Hilbert space (RKHS) in this paper. The reconstruction of a function in an RKHS from a given set of sampling points and the reproducing kernel of the RKHS is discussed. Firstly, we analyse and give the optimal approximation of any function belonging to the RKHS in detail. Then, a necessary and sufficient condition to perfectly reconstruct the function in the corresponding RKHS of complex-valued functions is investigated. Based on the derived results, another proof of the sampling theorem in the linear canonical transform (LCT) domain is given. Finally, the optimal approximation of any band-limited function in the LCT domain from infinite sampling points is also analysed and discussed.

Keywords

reproducing kernel Hilbert space (RKHS)sampling theoremlinear canonical transform (LCT)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 1 , February 2015 , pp. 109 - 120
Copyright
Copyright © Cambridge University Press 2014 

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Footnotes

1

This work was supported by the National Natural Science Foundation of China (No. 61171195), the Program for New Century Excellent Talents in University. (No. NCET-12-0042), and also supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 61421001).

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