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Under what conditions can you integrate and differentiate a Fourier series representing a real-valued function such that the result converges over some open subset of the domain?

Basically, if a periodic function has a convergent Fourier representation, I'd like to you know can integrate it and differentiate.

Please no circular answers like "if the integral is absolutely convergent..." because that doesn't tell us anything about how to work with the most minimal starting information.

StackQuest
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  • You have asked "under what conditions". If such a condition is absolute convergence, then that is the most concise answer! It is not in the least bit circular. What would be far more tricky would be fully characterising absolute convergence in an at-a-glance way for every possible Fourier series... I don't know much about the topic but I expect this is impossible – FShrike May 27 '22 at 17:00
  • If I say "when does it converge" and you say "when it converges" that's text-book circular logic, not the least bit helpful. – StackQuest May 27 '22 at 21:15

1 Answers1

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Of course, after you differentiate, the resulting series must converge. For example, $$ \sum_{k=1}^\infty \frac{e^{i k t}}{k} = \log(1-e^{it}) $$ converges on $(0,2\pi)$. [And it converges in the $L^2$ norm, since $\sum\frac{1}{k^2} < \infty$.]

But the derivative $$ \sum_{k=1}^\infty i e^{i k t} $$ diverges everywhere. [And it diverges in the $L^2$ norm, since $\sum 1 = \infty$.]

In a certain sense, the "natural setting" for the question is convergence in the mean of order $2$, rather than pointwise convergence.

GEdgar
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  • I'm looking for real-world applications to actual wave-forms that occur in physical reality. I don't get the sense that this isolated example meets those standards. Something must categorize real-valued wave forms to ensure convergence, obviously we can look at a real wave form and draw boxes under it that approximate its area, or calculate slopes locally, your specific example doesn't stop me from doing that, but your post doesn't touch on that in the slightest. – StackQuest May 27 '22 at 21:22
  • @StackQuest None of your comment here is present in your question, but it should be since it's all part of what you're looking for. Please update your question to contain all the relevant information :) – postmortes May 30 '22 at 05:26