An Abstraction of Shannon's Sampling Theorem

Ikuji HONDA  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E81-A   No.6   pp.1187-1193
Publication Date: 1998/06/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: General Fundamentals and Boundaries
Keyword: 
sampling theorem,  nonband-limited signal,  Riesz basis,   frame,  multiresolution analysis,  

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Summary: 
This paper proves a general sampling theorem, which is an extension of Shannon's classical theorem. Let o be a closed subspace of square integrable functions and call o a signal space. The main aim of this paper is giving a necessary and sufficient condition for unique existence of the sampling basis {Sn}o without band-limited assumption. Using the general sampling theorem we rigorously discuss a frequency domain treatment and a general signal space spanned by translations of a single function. Many known sampling theorems in signal spaces, which have applications for multiresolution analysis in wavelets theory are corollaries of the general sampling theorem.