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I've been reading Widder's Advanced Calculus text, which says that there are some continuous functions that have divergent Fourier series, which are summable to the function (C, 1).

I'd greatly appreciate it if someone could present an explicit example of such a function, I haven't been able to find one anywhere.

(When Widder mentions divergence of the Fourier series of a continuous function, is it actually possible for it to diverge everywhere, or can it only diverge "almost everywhere?")

Edit: I found something in Zygmund's book, but it's somewhat unclear. Clarification would be helpful, thank you!

Ayesha
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  • Well, you can construct one, but it is messy. The book "Fourier Analysis" by Körner provides with a construction of a continuous function whose Fourier series diverges at $0$. Kolmogorov once gave an example of a continuous functions whose Fourier series diverges everywhere. Maybe you can Google that. – Pedro Dec 15 '13 at 22:25
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    On a side note, every continuous $f$ function has $(C,1)$-summable Fourier series that converges to $f$. In fact, every integrable function has $(C,1)$-summable Fourier series that converges to $\dfrac{f(\cdot^{+})+f(\cdot^{-})}2$ whenever the limit exists. This is Fejer's theorem. – Pedro Dec 15 '13 at 22:27
  • I've tried Googling the Kolmogorov example, yes, but I didn't get much. – Ayesha Dec 15 '13 at 23:37
  • Pedro Tamaroff, when you say integrable, do you mean Riemann-integrable or Lebesgue-integrable? – Ayesha Dec 15 '13 at 23:38
  • See this question: http://math.stackexchange.com/questions/14855/an-example-of-a-continuous-function-whose-fourier-series-diverges-at-a-dense-set – Igor Rivin Dec 16 '13 at 00:49
  • @PedroTamaroff, What page on Korner's "Fourier analysis"? – RFZ Jan 10 '16 at 16:47

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