Summary form only given. The Frazier-Jawerth transform (FJT), originally the phi-transform, is similar to the wavelet transform and is distinguished by the fact that the analyzing functions form an overcomplete basis for he signal space and may be nonorthogonal. This added flexibility makes possible the definition of optimal analyzing functions, which are the focus of this study. For continuous-time and infinite discrete-time signals, the optimally localized functions are the prolate spheroidal wave functions and their discrete versions. For finite discrete-time signals and images, generalizations of these functions that are applicable for use in the FJT have been identified by the authors. >
A discrete time series has associated with it an amplitude spectrum which is a periodic function of frequency. This paper investigates the extent to which a time series can be concentrated on a finite index set and also have its spectrum concentrated on a subinterval of the fundamental period of the spectrum. Key to the analysis are certain sequences, called discrete prolate spheroidal sequences, and certain functions of frequency called discrete prolate spheroidal functions. Their mathematical properties are investigated in great detail, and many applications to signal analysis are pointed out.
Berzelius failed to make use of Faraday's electrochemical laws in his laborious determination of equivalent weights.