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Divergence

Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. For example, the Fourier series of a continuous T -periodic function need not converge pointwise. The uniform boundedness principle yields a simple non-constructive proof of this fact.

In 1922, Andrey Kolmogorov published an article titled Une série de Fourier-Lebesgue divergente presque partout in which he gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere. He later constructed an example of an integrable function whose Fourier series diverges everywhere.

It is possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function f defined for all x in by

Because the function is even the Fourier series contains only cosines:

The coefficients are:

As m increases, the coefficients will be positive and increasing until they reach a value of about C m ≈ 2 / (n 2 π) {\displaystyle C_{m}\approx 2/(n^{2}\pi)} at m = 2 n 3 / 2 {\displaystyle m=2^{n^{3}}/2} for some n and then become negative (starting with a value around − 2 / (n 2 π) {\displaystyle -2/(n^{2}\pi)}) and getting smaller, before starting a new such wave. At x = 0 {\displaystyle x=0} the Fourier series is simply the running sum of C m, {\displaystyle C_{m},} and this builds up to around

in the n th wave before returning to around zero, showing that the series does not converge at zero but reaches higher and higher peaks. Note that though the function is continuous, it is not differentiable.