On Gel'fand-Shilov's Theorem for Bandlimited Signals

Hiroshi SUGIYAMA  

IEICE TRANSACTIONS (1976-1990)   Vol.E73   No.5   pp.653-657
Publication Date: 1990/05/25
Online ISSN: 
Print ISSN: 0000-0000
Type of Manuscript: Special Section PAPER (Special Issue on Information Theory and Its Applications)

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All distributions in the sense of Schwartz are considered as signals and their Fourier transforms defined by Gel'fand and Shilov as frequency spectra. The necessary and sufficient condition for a signal to be bandlimited is that it is an entire function of exponential type. This follows from the Gel'fand and Shilov's theorem which states that any distribution whose Fourier transform has a compact support reduces to an entire function of exponential type and vice verse. But their original proof was incomplete; their definition of 'compact support' was insufficient. Here, a reasonable definition of 'compact support' is introduced and a rigorous proof of their theorem is presented. Once the theorem is established, many applications in various fields will be possible. Above all the so-called sampling theorem for bandlimited signal becomes the interpolation/extrapolation for entire function of exponential type. As it is shown from the Gel'fand and Shilov's theorem with the completion presented in this paper that the frequency hand of a bandlimited signal is a point set on the complex plane congruent with the complement of the existence domain of its Borel transform, an explicit interpolation/extrapolation formula with sampling points on the negative time axis can be constructed.