Fourier series | Real analysis | Numerical artefacts

Gibbs phenomenon

In mathematics, the Gibbs phenomenon, discovered by Henry Wilbraham and rediscovered by J. Willard Gibbs, is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The function's th partial Fourier series (formed by summing its lowest constituent sinusoids) produces large peaks around the jump which overshoot and undershoot the function's actual values. This approximation error approaches a limit of about 9% of the jump as more sinusoids are used, though the infinite Fourier series sum does eventually converge almost everywhere except the point of discontinuity. The Gibbs phenomenon was observed by experimental physicists, but was believed to be due to imperfections in the measuring apparatus, and it is one cause of ringing artifacts in signal processing. (Wikipedia).

Gibbs phenomenon
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