Riemann Lebesgues lemma in $L^p$

Question

Is it still true that :

$$ \int_{\mathbb R } [f(x) \sin(nx)]^p dx \to 0 \text{ when } n \to \infty $$

I can't find anything about it. Thanks.

Answer 1

It fails for $p=2$. Indeed, $$ \sin^2(nx)=\frac{e^{2inx}-2+e^{-2inx}}{-4}=\frac 12-\frac 12\cos\left(2nx\right) $$ hence for each function $f$ which is square integrable, in view of the classical Riemann Lebesgue lemma, $$ \int_{\mathbb R } [f(x) \sin(nx)]^2 dx \to 1/2 $$ and not $0$. The reasoning can be extended to the case where $p$ is an even integer.

Also, a version in $L^p$ with absolute values will fail for $p=1$ because $$ \left\lvert f(x)\sin(nx)\right\rvert\geqslant \left\lvert f(x)\right\rvert \sin^2(nx)=\left\lvert f(x)\right\rvert\left(\frac 12-\frac 12\cos\left(2nx\right)\right).$$