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One of the fundamental problems in communications is the transmission of a signal through a bandlimited channel. It should, therefore, be of great interest to find a general method of transmitting an arbitrarily close approximation to any finite signal through a channel which is arbitrarily bandlimited. The present paper develops such a method and evaluates the cost in time and energy to accomplish this feat. The problem of fitting an arbitrarily bandlimited signal to a finite number of arbitrary points is solved, and the minimum energy signal, fitting the points and having a spectrum confined to the given pass band, is found. The behavior of this signal, with shrinking bandwidth, is investigated.