Fourier integral and coefficients in given function

Question

I found this problem while studying and I can't find its solution.

The function is given $F: \mathbb{R} \to \mathbb{R}$ with this formula: $F=\alpha \sin(\beta w)/w$, $\alpha$ and $\beta$ both from $\mathbb{R}^+$.

1. determine $\alpha_0$ and $\beta_0$ from given graph
2. if function $F$ is the Fourier transform of certain function $f$, find explicit analytic form of function $f$ and draw it.

I don't know if I can use classic ways to find these coefficients like taking $F(0)$ or $F'(0)$ to determine them. Second part is about solving Fourier integral and basic function graphing I think.

Answer 1

HINT

(1) Usually, $\sin x$ has period $2\pi$. What is the period of $\sin(\beta x)$? Determine the period from the graph and this will tell you $\beta$. Also, what is $\lim_{w \to 0} F(w)$? This should tell you $\alpha$.

(2) You could do it by inverse Fourier transform as you suggest. What do you get when you try it?