Gibbs phenomenon

In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The ${\textstyle N}$^th partial Fourier series of the function (formed by summing the ${\textstyle N}$ lowest constituent sinusoids of the Fourier series of the function) produces large peaks around the jump which overshoot and undershoot the function values. As more sinusoids are used, this approximation error approaches a limit of about 9% of the jump, though the infinite Fourier series sum does eventually converge almost everywhere (pointwise convergence on continuous points) except points of discontinuity.^[1]

The Gibbs phenomenon was observed by experimental physicists and was believed to be due to imperfections in the measuring apparatus,^[2] but it is in fact a mathematical result. It is one cause of ringing artifacts in signal processing. It is named after Josiah Willard Gibbs.