Showing decay of Fourier coefficients $C_n = 1/2\pi \int_{-\pi}^\pi e^{-inx} \phi(x) dx$

Question

I'm looking at the Fourier coefficients of $\phi \in L^1([-\pi, \pi])$ defined as

$$ C_n = \frac{1}{2\pi} \int_{-\pi}^\pi e^{-inx} \phi(x) dx$$I want to show that $\lim_{|n| \to \infty} C_n = 0$ I can show that $\sup|C_n| \leq \frac{1}{2\pi}\|\phi\|_{L^1}$ using a trick similar to what was discovered in my earlier question here.

I don't know how to show that $C_n \to 0$ though.

Answer 1

This result is known as Riemann-Lebesgue lemma. Prove it in two steps:

1. Suppose that $\phi$ is $C^1$ (and periodic). Integration by parts shows that $C_n\to0$.
2. Use a density argument. Any $\phi\in L^1$ can be approximated in the $L^1$ norm by $C^1$ functions.