Fourier transform of a function with $f(x)=-f(-x)$

Question

If $f\in L^1(\mathbb R)$ with $f(x)=-f(-x)$ do we have $\lim_{\xi\to\infty} \hat{f}(\xi)=\hat{f}(0)$? I know that $\hat{f}(\xi)=-\hat{f}(-\xi)$ and that $\lim_{\xi\to\infty} \hat{f}(\xi)=0$.

Answer 1

Fourier transform of an odd function is odd and hence $\hat{f}(0)=0$. Then by Riemann-Lebesgue lemma Fourier transform of an $L^1(\Bbb R)$ is a member of $C_0(\Bbb R)$ and hence $\lim_{\xi \to \infty} \hat{f}(\xi)=0$.