It is wellknown that there is a convergence in norm for Fourier series in $L_p$, if $1<p<\infty$, but are there some examples for pointwise divergence if $p=1,\infty$ in books, or somewhere? I have only found Kolmogorov example, but it is too complicated, and i don't need divergence almost everywhere
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$\begingroup$ There are continuous functions with Fourier series divergent at a point, so your $L^p$ assumptions seems somewhat beside the point here. (Also, since you talk about Fourier series, presumably you're on the circle, so $L^{\infty}\subset L^p$.) $\endgroup$– Christian RemlingCommented May 19, 2015 at 21:59
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2 Answers
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Any function that is in $L^1$ but not in $L^\infty$ will have some point in whose neighbourhood it is unbounded, and the Fourier series is likely to diverge at such a point. A simple example is $$ f(x) = \ln(1-\cos(x)) = -\ln(2) + \sum_{n=1}^\infty \dfrac{2 \cos(nx)}{n} $$ where the series diverges at multiples of $2\pi$.
EDIT: For Fejér's example of a continuous function whose series diverges at a point, see e.g. Edwards, "Fourier series", sec. 10.3.1.
answered May 19, 2015 at 20:13
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$\begingroup$ But why $ln(1-cos(x))$ is in $L_1$, why is it integrable? $\endgroup$ Commented May 19, 2015 at 20:45
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$\begingroup$ As $x \to 0$, $\ln(1-\cos(x)) \sim 2\ln|x|$, which is an integrable singularity. $\endgroup$ Commented May 20, 2015 at 0:21
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$\begingroup$ Thanks! But I also can't understand why does Uniform Boundedness Principle implies divergence of Fourier series in $L_\infty$, I have read it somewhere, but it seems not to be so obvious for me $\endgroup$ Commented May 20, 2015 at 8:31
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$\begingroup$ @user228494: Uniform boundedness principle. An interesting method to show existence. But your question asks for "some examples" and UBP is not the way to produce explicit examples. $\endgroup$ Commented May 20, 2015 at 12:57
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See Katznelson's Harmonic analysis (page 61) for a very simple example of a continuous function whose fourier series diverges at a dense set of points. I believe du Bois Reymond is the person responsible for the first such example.
answered May 20, 2015 at 3:47