Characterization of Periodic Spline Functions in Comparison with Fourier Series


IEICE TRANSACTIONS (1976-1990)   Vol.E72   No.6   pp.702-709
Publication Date: 1989/06/25
Online ISSN: 
Print ISSN: 0000-0000
Type of Manuscript: PAPER
Category: Numerical Calculation and Mathematical Programming

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Spline functions attract attention as useful approximate functions which are smooth and apt not to vibrate. But the approximation by them is not characterized with any physical meanings such as the harmonic frequency for the Fourier series. The present paper aims to analyze characteristics of the spline approximation in the aspect of comparison with those of the Fourier series approximation. The results are summarized as follows. (1) Characteristics of the spline approximation change from those of the staircase approximation toward those of the Fourier series approximation according to the increase of the order of the spline function. This means that the spline approximation can adjust its characteristics to a signal given to be approximated if we choose an appropriate order. (2) The approximation power of a spline function is not constant on the time axis while that of a Fourier series is constant. A spline function has the spans with higher approximation power and those with lower power in turn on the time axis. This means that better approximation can be obtained by putting the spans with higher approximation power at the spans where good approximation is desired such as at peaks of a given signal. The above findings contribute to better use of spline functions in approximating signals.