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 function's Nth partial Fourier series (formed by summing its N 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.

In practice, the difficulties associated with the Gibbs phenomenon can be ameliorated by using a smoother method of Fourier series summation, such as Fejér summation or Riesz summation, or by using sigma-approximation.
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