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Gibbs Phenomena for 1D Fourier Series

The pointwise limit of the Fourier series of a function with discontinuities does not converge pointwise to the original function everywhere.

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number of series terms — number of terms taken into account for the Fourier series expansion
The Fourier series of a continuous, sufficiently smooth function converges pointwise to the original function. For discontinuous functions, the series converges in the norm but does not converge pointwise. At points where the function is discontinuous, the value of the series can deviate from the original function value; at a jump discontinuity, the value of the Fourier series converges to the average of the values of the left-and right-sided limits of the original function. For truncated Fourier series, the partial sums will overshoot the original function value for values near discontinuities; this is known as the Gibbs phenomenon.
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