Example 1. Find the function $f(x)$, the sine Fourier transform of which is

$$
F(p)=\left[\begin{array}{ll}
1, & 0<p<1 \\
0, & 1<p<\infty
\end{array}\right.
$$
## Solution.

1. Check that condition (1) is satisfied:

$$
\int_{0}^{\infty}|F(p)|^{2} d p=\int_{0}^{1} d p=1<\infty
$$

2. Evaluating the integral on the right-hand side of (2), we get:

$$
f(x)=\sqrt{\frac{2}{\pi}} \int_{0}^{\infty} F(p) \sin p x d p=\sqrt{\frac{2}{\pi}} \int_{0}^{1} \sin p x d p=\sqrt{\frac{2}{\pi}} \frac{1-\cos x}{x}
$$

Answer. $f(x)=\sqrt{\frac{2}{\pi}} \frac{1-\cos x}{x}$.