I don't know if this is a very simple question with a very difficult answer, but:
Is $y = \dfrac{\sin x}{x}$ the only function such that
$\int_{-\infty}^{\infty} f(x) dx =\int_{-\infty}^{\infty} f^2(x) dx \text{ ?}$
This is to say, if we start with the integral equation
$\int_{-\infty}^{\infty} f(x) dx =\int_{-\infty}^{\infty} f^2(x) dx \text{ ?}$
would we only find $f(x) = \dfrac{\sin x}{x}$, or we can expect other solutions?
Maybe I should have made this clear, but I'm talking about $f(x) \neq 0$ and $f(x)$ continuous in $\mathbb{R}$.