Prove that $ \lim_{n \to \infty} \int_{-\infty}^{\infty} \sin(nt) f(t) d t = 0 $.

Question

I am trying to prove that $ \lim_{n \to \infty} \int_{-\infty}^{\infty} \sin(nt) f(t) d t = 0 $ for every Lebesgue integrable function $ f $ on $ \mathbb{R} $. My first thoughts were to use Dominated Convergence Theorem but I realised that there is no pointwise limit of the sequence of functions $ f_n = \sin(nt) f(t) $. I do not know how to proceed.

Any help would be appreciated. Thanks!

Answer 1

(This is the content of the Riemann-Lebesgue Lemma.)

Sketch of proof:

1. Fix $\epsilon > 0$, there exists $g$ that is continuously differentiable and has compact support, such that $\|f - g\|_{L^1} < \epsilon / 2$.
2. $\int \sin(nt) g(t) dt = \frac{1}{n} \int \cos(nt) g'(t) dt \to_{n\to\infty} 0$.
3. Hence there exists $N > 0$ such that for every $n > N$, $|\int \sin(nt) f(t) dt| < \epsilon$.