Study of the superiority of generalized linear interpolation approximation
Takuro Kida, Hiroshi Mochizuki · Electronics and Communications in Japan (Part III Fundamental Electronic Science) · 1991
Abstract Consider a set of waveforms whose Fourier spectra are band‐limited within a certain frequency band. We assume that the weighted Lp integrals (1<p< +∞) of the Fourier spectra are bounded with respect to a given weight function. We present a method which optimally approximates the original waveform contained in the above set with respect to an appropriate measure of error. We use a series of sample values of the outputs obtained when the waveform passes through a given finite number of linear time‐invariant filters. Here, we establish a theorem which shows that there exists a waveform among all those waveforms whose sample values at the sample points are all zero, whose error is a maximum at some arbitrary fixed time t. Further, under the condition that p = 2 and the sample values, which are in general not all zero, are given, we prove that the upper limit of approximation error is proportional to the maximum of absolute value of the approximation error described above. As a result, it is shown that as long as the set of waveforms and the measure of approximation error mentioned above are adopted, the proposed linear approximation method is superior to all other approximation methods even including, for example, non‐linear signal processing using the same sample values. Further, at the end of this paper, we show that our argument can be applied to more general sets of waveforms.