Gibbs Phenomenon
The non-uniform convergence of the Fourier series for discontinuous functions is known as the Gibbs phenomenon. In 1899 American mathematician Josiah Willard Gibbs noticed that near a point where a function has a jump discontinuity, the partial sums of a Fourier series show a substantial overshoot near these endpoints. Carrying out the sums of the Fourier series to a higher number of terms will not diminish the amplitude of the overshoot although the overshoot occurs over a smaller and smaller interval. This overshoot exhibits itself in an oscillatory behavior near the discontinuous point(s) of the function. Although Wilbraham first analyzed this phenomenon in 1848, it was Gibbs that studied it in detail and for whom the behavior is named. Later, in 1906 Bôchner generalized this phenomenon to arbitrary functions. The Gibbs phenomenon is not only observed in Fourier series but also occurring at simple discontinuities in other eigenfunction series.
The magnitude of the overshoot is dependent upon the type of discontinuity and not on the values of the function studied. Normally the overshoot is about 9% of the magnitude of the discontinuity jump but it is sometimes as high as 17.9%. Gibbs showed that is a function of x (f(x)) is piecewise smooth on [-, ], and x0 is a discontinuity point, then the Fourier partial sums will exhibit a overshoot with height almost equal to 0.09(f(x0max)-f(x0min)), where f(x0max) is the maximum value of the function at the point of discontinuity and f(x0min) is the minimum value of the function at that point.
The Wilbraham-Gibbs constant, usually denoted G, actually quantifies the degree to which the Fourier series of a function overshoots the function value at a discontinuity. This constant is useful but the reader needs to be aware that there are differences in its usage. This constant is not really a constant. Sometimes the limiting crest of the highest wave deviating from the actual function is denoted by 2G where G = 0 sin()/ d = 1.851937, whereas other times it may be denoted as 1/2+G/ = 1.089489, and still other times it is denoted as 2G/ = 1.178979.
There are complex methods to smooth the Gibbs phenomenon. One method is called the -approximation or sometimes it is referred to as the Lanczos sigma factor. In this approximation a function is multiplied by the coefficients in the Fourier partial sums. This function is a complex sin function involving the period of the original function. Another form of smoothing is called Hanning smoothing. Physical scientists who often observe a complex phenomenon using a large bandwidth detection system to collect as much information as possible in the shortest amount of time usually employ this type of smoothing. The technique involves reducing the number of data points around a discontinuity in the raw data before the Fourier transform is taken. Although this smoothes the Gibbs phenomenon it also degrades the frequency resolution.
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