I need functional $F(f,g)$ to measure similarity of two real-valued functions $f$, $g$ defined on real segment [a, b]
As a simple reasonable example we can consider functional
$F(f,g) = \frac{1}{b-a}\int_a^b e^{-(f(x)-g(x))^2} dx$
which have desired properties $F(f,f) = 1$; $F(f,g)\lt 1$
I wandering about a bit sophisticated $F$ that can reveal similarity of two function if:
- $g(x) = \alpha f(x)$
- $g(x) = f(\alpha x)$
- $g(x) = \theta (x-a_1) \theta (b_1-x) f(x)$ ($g$ is only 'part' of function $f$)
Maybe problem is simpler if functions $f,g$ are defined on finite set ${x_i}$