I am trying to understand the infinite dimensional space such as $L^2$ space \begin{align} \langle f,g\rangle = \int f(x) \bar{g}(x)dx. \end{align} where $\bar{g}$ denote the conjugate of $g$. I have this question when I am trying to understand the Fourier transform in a more general framework. We know the Fourier transform is given by \begin{align} F(w)=\langle f(x),g_w(x)\rangle = \int f(x)\bar{g}_w(x)dx, \end{align} where $g_w(x) = e^{j2\pi wx}$.
I want to generalize the Fourier transform to the case with arbitrary $g_w(x)$. From my understanding, $F(w)$ can be treated as the probjection on the function $g_w(x)$.
Similary as the Fourier function, the only condition I impose on the $g_w(x)$ is the orthogonality. Specifically, we assume that $\langle g_{w_1}(x),g_{w_2}(x) \rangle = 0, \forall w_1 \neq w_2$. Thus, $\{g_{w}(x): w \in \mathbb{R}\}$ forms the orthogonal basis of the function, not necessarily the Fourier function.
Considering the intuition of projection and basis, then we can have something like that \begin{align} f(x) = \sum_w F(w)g_w(x) \end{align} The diffenrence is that the number of basis function $\{g_{w}(x): w \in \mathbb{R}\}$ is infinite. We should have integration instead of summation, \begin{align} f(x) = C\int \langle f,g_{w} \rangle g_w(x) dw ~~~~~~~~~~(1) \end{align} where $C$ is some constant. This equation (1) is just my intuition not a strict proof.
My question is that do we have some theorem or lemma which support the equation (1) when the orthogonality holds for $g_w(x)?$ By the way, I know (1) holds for Fourier function, and here I am focusing on arbitrary basis function $g_w(x)$.