Page "Gibbs phenomenon" Paragraph 18

The Gibbs phenomenon is also closely related to the principle that the decay of the Fourier coefficients of a function at infinity is controlled by the smoothness of that function ; very smooth functions will have very rapidly decaying Fourier coefficients ( resulting in the rapid convergence of the Fourier series ), whereas discontinuous functions will have very slowly decaying Fourier coefficients ( causing the Fourier series to converge very slowly ).

Note for instance that the Fourier coefficients 1, − 1 / 3, 1 / 5, ... of the discontinuous square wave described above decay only as fast as the harmonic series, which is not absolutely convergent ; indeed, the above Fourier series turns out to be only conditionally convergent for almost every value of x.

This provides a partial explanation of the Gibbs phenomenon, since Fourier series with absolutely convergent Fourier coefficients would be uniformly convergent by the Weierstrass M-test and would thus be unable to exhibit the above oscillatory behavior.

By the same token, it