fa.functional analysis - Generalize Fourier transform to other basis than trigonometric function

You need the orthogonality condition to get such an integral representation for the coefficients; otherwise it would probably be more complicated.

The Fourier series of any $L^2$ function converges not only in the norm (which follows from the fact that ${e^{inx}}$ is an orthonormal basis) but also almost everywhere (the Carleson-Hunt theorem). Both these assertions are also true in any $L^p,p>1$ but at least the first one requires different methods than Hilbert space ones. In $L^1$, by contrast, a function's Fourier series may diverge everywhere.

There are many conditions that describe when a function's Fourier series converges to the appropriate value at a given point (e.g. having a derivative at that point should be sufficient). Simple continuity is insufficient; one can construct continuous functions whose Fourier series diverge at a dense $G_{delta}$. The problem arises because the Dirichlet kernels that one convolves with the given function to get the Fourier partial sums at each point are not bounded in $L^1$ (while by contrast, the Fejer kernels or Abel kernels related respectively to Cesaro and Abel summation are, and consequently it is much easier to show that the Fourier series of an $L^1$ function can be summed to the appropriate value using either of those methods). Zygmund's book Trigonometric Series contains plenty of such results.

There is a version of the Carleson-Hunt theorem for the Fourier transform as well.