Gibbs Phenomenon - Formal Mathematical Description of The Phenomenon

Formal Mathematical Description of The Phenomenon

Let be a piecewise continuously differentiable function which is periodic with some period . Suppose that at some point, the left limit and right limit of the function differ by a non-zero gap :

For each positive integer N ≥ 1, let SN f be the Nth partial Fourier series

S_N f(x) := \sum_{-N \leq n \leq N} \hat f(n) e^{2\pi i n x/L} = \frac{1}{2} a_0 + \sum_{n=1}^N \left( a_n \cos\left(\frac{2\pi nx}{L}\right) + b_n \sin\left(\frac{2\pi nx}{L}\right) \right),

where the Fourier coefficients are given by the usual formulae

Then we have

and

but

More generally, if is any sequence of real numbers which converges to as, and if the gap a is positive then

and

If instead the gap a is negative, one needs to interchange limit superior with limit inferior, and also interchange the ≤ and ≥ signs, in the above two inequalities.

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