General sampling theorems for functions in reproducing kernel Hilbert spaces
 M. Zuhair Nashed,
 Gilbert G. Walter
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Abstract
In this paper we prove general sampling theorems for functions belonging to a reproducing kernel Hilbert space (RKHS) which is also a closed subspace of a particular Sobolev space. We present details of this approach as applied to the standard sampling theory and its extension to nonuniform sampling. The general theory for orthogonal sampling sequences and nonorthogonal sampling sequences is developed. Our approach includes as concrete cases many recent extensions, for example, those based on the SturmLiouville transforms, Jacobi transforms, Laguerre transforms, Hankel transforms, prolate spherical transforms, etc., finiteorder sampling theorems, as well as new sampling theorems obtained by specific choices of the RKHS. In particular, our setting includes nonorthogonal sampling sequences based on the theory of frames. The setting and approach enable us to consider various types of errors (truncation, aliasing, jitter, and amplitude error) in the same general context.
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 Title
 General sampling theorems for functions in reproducing kernel Hilbert spaces
 Journal

Mathematics of Control, Signals and Systems
Volume 4, Issue 4 , pp 363390
 Cover Date
 19911201
 DOI
 10.1007/BF02570568
 Print ISSN
 09324194
 Online ISSN
 1435568X
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Sampling theorems
 Reproducing kernels
 Bandlimited signals
 Nonuniform sampling
 Nonorthogonal sampling sequences
 Frames
 Sampling errors
 Authors

 M. Zuhair Nashed ^{(1)}
 Gilbert G. Walter ^{(2)}
 Author Affiliations

 1. Department of Mathematical Sciences, University of Delaware, 19716, Newark, Delaware, U.S.A.
 2. Department of Mathematical Sciences, University of Wisconsin, 53201, Milwaukee, Wisconsin, U.S.A.