$L^2$ - base functions say we have a function, $f$, in $L^2$ and have different base functions for $L^2$, then is there a reason to believe that one base is "better" compared to the other when writing expansions for $f$? Could it be that for a particular $f$ a specific base function provides a faster convergence of the corresponding finite expansions?
Many thanks for your hints.
 A: If "better" means better suited for certain purposes then yes. Fourier Series, for example, allow approximation which are extremely well suited for application in physics in general, also for numerical approximation (fast Fourier transform) or for analysing partial differential equations (since they map differential operators to multiplicatin operators). It also turns out that Fourier Series behave 'not too bad' when approximating functions in other spaces, like $C^0$.
Polynomial based ONBs are often useful for other ODEs. In higher dimensions spherical harmonics have similar advantages.
Wavelets are, as it seems, a good choice for audio compression and yield other information than Fourier analyis. These are just two examples which I often came across, so the came to mind immediately. I'm sure there are many others.
When you are analyzing linear operators this is often easier when you can find on ONB of eigenvectors (this is a general Hilbert space example, but $L^2$ is a typical model for Hilbert spaces).
