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Bounded Determinable Subsets of Banach Spaces

Balram S. Rajput
SIAM Journal on Applied Mathematics
Vol. 20, No. 4 (Jun., 1971), pp. 735-748
Stable URL: http://www.jstor.org/stable/2099871
Page Count: 14
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Bounded Determinable Subsets of Banach Spaces
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Abstract

Root [5] has inaugurated a study in communication theory which is concerned with the approximate determination of the characteristics of channels by transmitting a known test signal and performing a finite number of measurements on the output of the channel. In the spirit of Root's work, Prosser and Root [4] further investigated this problem. Here a channel is described as a Hilbert-Schmidt operator on signal Hilbert space L2(R) into itself (no random disturbances are involved). The problem treated is the approximate determination of channels, which are known to belong to a bounded subset D of the Hilbert space of all Hilbert-Schmidt operators on L2(R). The determination is made through a finite number of measurements of the channel outputs due to an optimally selected input test signal. The approximation is made in the mean square sense. One of the important results is the characterization of the sets D for which such identification is possible. In the present work, a channel is described by an arbitrary linear continuous operator from a real signal Banach space B into itself. Apart from other results, we are able to characterize: (a) the bounded subsets D of B for which it is possible to distinguish every signal to within a given error, defined in terms of the norm, by a finite number of measurements; (b) the bounded, convex and circled subsets D of L(B), the real Banach space of all bounded linear operators on B, for which it is possible to distinguish each element in the class to within a given error, defined in terms of the norm, by introducing one or a finite number of optimally selected input signals and making a finite number of measurements on the output signals.

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