Does anyone know how the graph looks like for a continuous function with Fourier series diverging at $x=0$ ?

The example due to Fejer (a variation of the du Bois-Reymond construction), is explicitly given by $$f(x)=\sum_{n=1}^\infty \frac1{n^2} \sin(2x2^{n^3})\big[\sum_{m=1}^{ 2^{n^3}} \frac{sin(mx)}m \big]$$ see eg Champeney, Handbook of Fourier Theorems, page 37. However, trying to draw $f(x)$ with basic software seems too slow, even for a few terms in the series...

Does anyone know a place where one such graph is shown?

Of course, by the Dini and Jordan convergence tests, the function will have essentially no smoothness at $x=0$, and won't be of bounded variation (so can expect lots of oscillations around $x=0$). Also, to guess its real graph one should draw the Cesaro sum of the series (or some other good approximation), as the Fourier series will show the divergent behavior stated by the theorem.

  • $\begingroup$ Nina Bary commented in <I>A treatise on trigonometric series volume 1</I>, page 128 that it is impossible to represent Fejer's example (page 127) by curve and to explain geometrically the divergence of the series at 0. She gave Lebesgue's example in section 46 (page 128 -133) and a sketch of Lebesgue's example and an explanation for its divergence at 0. $\endgroup$ – user231543 May 3 '15 at 18:44
  • $\begingroup$ Nina added that the divergence of the Fourier series at 0 is a consequence of the fact that the Lebegsue's example is of unbounded variation in a neighbourhood of 0. $\endgroup$ – user231543 May 3 '15 at 18:55

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