We all know for any kind of signal $f(x)$, it corresponds a Fourier transform function $F(\omega)$ that $$ f(t) = \frac{1}{2\omega} \int_{-\infty}^{+\infty} F(\omega) e^{i\omega t} d\omega $$
So, $F(\omega)$ can be recognized as the component of $f(t)$ in $e^{i \omega t}$. This is similar to orthogonal analysis. Like a vector $a = (1,2,3)$ can be represented as $$ a= 1x+2y+3z $$ where $x$,$y$,$z$ are the orthonormal basis.
However, these two things are different in one perspective.Among $x$,$y$,$z$, there is no concept as 'distance'. For example, in Fourier transform. we can say that the distance between $\omega_1 = 1$ and $\omega=2$ is smaller than the distance between $\omega_2$ and $\omega_3 =4$.
This algebra structure in Fourier transform definitely has some extra meaning, but what is it? Does it make Fourier transform more sense compared with normal spectrum analysis?
If you break everything into intervals, then you do have a proper orthogonal analysis. For example, $$ f(t)=\sum_{n=-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}\int_{(n-1/2)\delta}^{(n+1/2)\delta}\hat{f}(\omega)e^{i\omega t}d\omega = \sum_{n=-\infty}^{\infty}f_n(t). $$ In this case, $$ (f_n,f_m)=\int_{-\infty}^{\infty}f_n(t)\overline{f_m(t)}dt=0,\;\;\; n \ne m. \\ \|f\|^2=\sum_{n=-\infty}^{\infty}\|f_n\|^2 $$ If you similarly write $g=\sum_{n=-\infty}^{\infty}g_n(t)$ then $(g_n,g_m)=0$ for $n\ne m$, and $$ (f,g) = \sum_{n=-\infty}^{\infty}(f_n,g_n). $$ It is the Parseval identity that gives you this orthogonality from the following orthogonal decomposition: $$ \hat{f}(\omega) = \sum_{n=-\infty}^{\infty}\chi_{[(n-1/2)\delta,(n+1/2)\delta]}(\omega)\hat{f}(\omega) $$ So, if you're willing to work in a situation where you cannot resolve wavelengths down to infinite precision, you end up with discrete orthogonal components. And this type of discretization has a natural connection with integration. Wave packets formed from disjoint intervals of frequency are orthogonal. The basic building blocks have the form \begin{align} (\chi_{[(n-1/2)\delta,(n+1/2)\delta]})^{\vee} & = \frac{1}{\sqrt{2\pi}}\int_{(n-1/2)\delta}^{(n+1/2)\delta}e^{i\omega t}d\omega \\ & = e^{in\delta t}\frac{1}{\sqrt{2\pi}}\int_{(-1/2)\delta}^{(1/2)\delta} e^{i\omega t}d\omega \\ & = e^{in\delta t}\frac{1}{\sqrt{2\pi}}\frac{\sin(\delta\omega t/2)}{t} \end{align} All of these inidividual sinc functions are orthogonal, and, once $\delta$ is resolved down your measurement abilities, you have a countable orthogonal basis, at least practically speaking. They're independent, uncorrelated, orthogonal.
This type of approach works in a very general sense when dealing with ODEs and PDEs coming out of symmetric Physical systems. It's just that the abstract transform variable may represent different physical things such as wavelength, energy, momentum, position, etc.. In these settings, continuous spectrum generally can be approximated by discrete spectrum. And, practically speaking, at some resolution you can't tell the difference.