Behavior of fourier transformation

Question

This is my first question here and english is not my native language, so i hope everything i write will be fine.

Im interested in the behavior of a fourier transformation $\hat{f}(\xi)$ for $\xi\rightarrow\infty$ if i know that $f(x)\in o(|g(x)|)$ for $x\rightarrow 0$ and we know that $g$ tends to $0$ . For example $g(x)=x$, or $g(x)=1-exp(x)$.

I just found something for Laplace transformation but not fourier transformation.

I'm highly interested in any theorems that give some information about the behavior of the fourier transform at infinity if i know something about the behavior of the original function at zero.

Thanks for any answer!

Answer 1

Somewhat related to your question is the Riemann-Lebesgue lemma which states that $\lim_\limits{n\to{\infty}}\hat{f_n}\to0$. Even the fourier inversion, plancharel,bessel and parseval theorems can give you a lot of information about the nature of fourier coefficients.