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What is nature's error criterion?

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1 Author(s)
Ernst A. Guillemin ; Dept. of Electrical Engineering Massachusetts Institute of Technology Cambridge 39, Massachusetts

It is well known that the Fourier series is not the only trigonometric polynomial that may be used to represent a periodic function. It is a polynomial with the property that the mean square error between a partial sum and the given function is a minimum; that is to say, it approximates the given function so as to make the mean square error a minimum. This error criterion is only one of many that could be stipulated as fixing the manner in which the polynomial approximates the given function, and from a practical standpoint it isn't even a good one for many applications because it suffers from the Gibbs phenomenon. A Tschebyscheff-like approximation or the one inherent in the Cesaro sum which converges uniformly even at points of discontinuity may be preferable in many cases.

Published in:

IRE Transactions on Circuit Theory  (Volume:1 ,  Issue: 1 )