If we establish a set of orthogonal functions with resonant and decay characteristics quite similar to those of seismic signals, then a few, or at most a few dozen, terms of the series may match significant phases of seismic events. A few hundred terms of a Fourier series might be required for an equivalent match of these same events. One such set of orthogonal functions are those based on Laguerre polynomials.
This paper describes the Laguerre theory which leads to better Fourier transforms (spectra). Laguerre functions are defined over all time, and their frequency responses are known over all frequencies. Therefore, if the Laguerre series converges quickly, then only a few points in time determine the spectra over the entire frequency range. Conversely, a few points in frequency can determine a transient response over all time.
Seismic applications include the transient calibration of seismometers yielding magnifications and phase responses over all frequencies, the automatic Laguerre analysis of Rayleigh waves by an analog computer, and the use of the Laguerre filter as a correlator to enhance pP phases on seismograms.