I've always seen the inner product $\langle f(x), g(x)\rangle$ written as
$$\langle f(x), g(x)\rangle = \int_0^1 f(x)g(x)w(x) \ dx$$
where $w(x)$ is a weight function.
But why do we always integrate from $0$ to $1$? And why not other values?
If we do integrate over other values, why is integrating over the domain from $0$ to $1$ so common?
I would greatly appreciate it if people could please take the time to clarify this.
Best Answer
It's probably just a function of the material. The eigenfunctions are typically just orthogonal between $a$ and $b$ i.e.
$$ \int_{a}^{b} \phi_{n}(x)\phi_{m}(x) \sigma(x) dx =0 , \lambda_{n} \neq \lambda_{m} $$
typically in PDEs you start off with $0$ to $L$ i.e $$ \int_{0}^{L} \phi_{n}(x) \phi_{m}(x) \sigma(x) dx $$ and $\sigma(x) =1$ , $ \lambda_{n} = \left( \frac{n \pi }{L} \right)^{2} $ then $ \phi_{n}(x) = \sin(\frac{ n \pi x}{L})$ for instance
when we have $$ \frac{d^{2}\phi}{dx^{2}} + \lambda \phi =0 \\ \phi(0) = 0 \\ \phi(L) = 0$$