Gibbs Phenomenon

Updated 01 Oct 2026

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What it is
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  persistent overshoot near a jump disconintuity when approximating a function with truncated Fourier sums
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Behavior
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  increasing the number of terms narrows the oscillatory region, but the limiting overshoot is about $8.95%$ of the jump size.
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  for a jump from $-1$ to $1$ , this corresponds to a limiting peak near $1.17898$ .
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  at a jump, the Fourier series converges to the midpoint of the one-sided limits under the usual convergence conditions.
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Distinctions
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  the $8.95%$ value is a limiting result, not an exact peak for every finite truncation.
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  a continuous triangular wave does not have the same jump-discontinuity overshoot; its coefficients decay faster.
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Mitigation
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  weighted summation/windowing can reduce ringing whil broadening the transition
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Related
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  Fourier Series
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Linked references 2