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Let $f$ be integrable function. I wanted to prove that $\lim_{s \to \infty}\int_0^{2\pi}f(t)\sin(st)dt=0$.

Firstly I wanted to do it for every simple function. I started with intervals. When $f$ is a characteristic function of interval, the result follows easily.

Then my textbook says that from this result follow for every simple functions and I can't see it.

I wanted to prove it by pi-lambda theorem by considering set of intervals as $\pi$ system and {$A \subset [0,2\pi] : \lim_{s \to \infty}\int_Af(t)\sin(st)dt=0$} as lambda system, however I can't prove it is lambda system. More specifically I can't prove third axiom of lambda system, the one with the union of increasing family of sets.

It would be true if $\lim_{s \to \infty} \sum_{i=1}^\infty \int_{A_{i+1}-A_i}\sin(st)dt=0$ and this would be true if I could change order of sum and limit but I can't see why I can.

Is this result even true? Should I try it in different way?

Arturo Magidin
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romperextremeabuser
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  • For every interval $[a,b]$, we have $\int_a^b\sin(st),dt\to 0$ as $s\to\infty$ by direct evaluation. From here there are standard analysis arguments (density and stuff; I haven’t fully thought about the $\pi$-$\lambda$ approach for here, but this also seems reasonable). Other approaches include: proving it first when $f$ is $C^1$ (or even $C^{\infty}$) using integration-by-parts once, then using a density result. There’s also another ‘trick’ which is essentially how Stein-Shakarchi do it, and that is to exploit continuity of translations in $L^p$. – peek-a-boo Mar 23 '24 at 12:33

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