I was playing around with this question in my graphic calculator and noticed, that $\int_{-1}^1\sin(n\pi x)f(x)dx\to0$ for $n\to\infty$, where $f\in L^p((-1,1))$ and $1<p<\infty$. While trying to proof this I had the idea of writing $\int_{-1}^1\sin(n\pi x)f(x)dx=\sum_{i=-n}^{n-1}\int_{i/n}^{(i+1)/n}\sin(n\pi x)f(x)dx$, hoping it would simplify the question, but I can't see how this could help. Help proving or disproving this would be very appreciated.
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Set up the Riemann sum and carefully choose your partition points so that it's always zero. – CyclotomicField Jun 30 '23 at 22:08
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4Am I missing something? Isn't this just the famous fact that fourier coefficients go to $0$ as $n \to \infty$ (for well behaved enough functions)? – Charles Hudgins Jun 30 '23 at 22:14
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Yes @CharlesHudgins, but he probably does not know about that already so he must prove it – julio_es_sui_glace Jun 30 '23 at 22:17
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@JulesBesson I think I would be just as confused as Charles. It's very strange for someone to have a working knowledge of $L^p$ spaces before Fourier series. – Ninad Munshi Jun 30 '23 at 22:18
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This was the case for me haha – julio_es_sui_glace Jun 30 '23 at 22:19
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I do know about the Riemann-Lebesgue-Lemma if you mean that, but im not too familiar with, so should I just take the Fourier series of $\sin(n\pi x)f(x)$? – algebrah Jun 30 '23 at 22:19
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no, your integral is proportional to $b_n$ (the sine part) – julio_es_sui_glace Jun 30 '23 at 22:20
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@algebrah I think you are confusing the Fourier transform with Fourier series. Have you studied Fourier series? – Charles Hudgins Jun 30 '23 at 22:23
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I startet studying the Fourier series last week, but in a more abstract setting. I found how to calculate the the $b_k$'s on Wikipedia. Knowing about the Riemand-Lebesgue-Lemma, this should do it, thanks ^^ – algebrah Jun 30 '23 at 22:29
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That is a well known behavior of intgegration with rapidly oscillating periodic functions. – Mittens Jun 30 '23 at 22:29
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Pretend $f$ is $C^1$ and integrate by parts. Then …. – Ted Shifrin Jul 01 '23 at 01:40