I am reading the book analysis 1 from Otto Forster. In a previous chapter we have looked at taylor series. In both situations we look at a series of functions $f_n$. In Taylor-series we use the term: The taylor series $(f_n)$ converges uniformly to a function $f$. In Fourier series we look at a different kind of convergence. We say that the Fourier series $f_n$ converges in the quadratic mean to another function $f$ (Definition below). I understand that if a series of smooth functions converges uniformly to a function $f$ then this function $f$ must be also smooth. With fourier series, i.e. with quadratic-mean-convergence we have a different relation, namely that the bessel-inequality of a function $f$(Definition below) becomes an equality.
The purpose of my question is to get a better understanding of "what happens" when a fourier series converges to the quadratic mean among other things that involves to understand what is so special if the bessel-inquality becomes an equality.
Definitions:
(Notation: A series $\sum_{k=-\infty}^{\infty}a_k$ is the sequence of partial sums $\sum_{k=-n}^na_k,n\in\mathbb{N}$)
When dealing with fourier series we always look at funtions $f:\mathbb{R}\rightarrow \mathbb{C}$ which are (for the sake of simplicity) periodic such that $f(x+2\pi)=f(x)$ and which are integrable over the interval $[0,2\pi]$. The fourier series of the function is given with $\mathscr{F}[f](x):=\sum_{k=-\infty}^{\infty}c_ke^{ikx}$ where $c_k:=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-ikx}dx,k\in\mathbb{Z}$.
For such a function we always know that $\mathbb{}$ the bessel-inequality holds:(the sum on the LHS is well-defined because it always converges in given circumstances)
$\sum_{k=-\infty}^{\infty}|c_k|^2\leq \frac{1}{2\pi}\int_{0}^{2\pi}|f(x)|^2dx$
Definition of quadratic-mean-convergence, we look at a functions $f$ and $f_n,(n\in\mathbb{N})$ which are (for the sake of simplicity) periodic such that $f(x+2\pi)=f(x)$ and which are integrable over the interval $[0,2\pi]$. We define $||f||_2:=\frac{1}{2\pi}\int_{0}^{2\pi}|f(x)|^2dx\geq 0$. We say that the sequence of functions $f_n$ converge in quadratic-mean to $f$ if
$\lim_{n\rightarrow\infty}||f-f_n||_2=0$. By defintition we can say now that the fourier series of $f$ converges in quadratic-mean to $f$ iff the bessel-inequality becomes an equality.
I hope someone can give me some insight on the consequences of this result. I have unterstood that in the case of fourier-series that quadratic-mean convergence is more general than uniform-convergence. Because if $(f_n)$ converges uniformly to $f$ then also it does in quadratic-mean. The book just says that the converse is not true. So lets suppose we have a fourier series that converges to the quadratic mean to a function $f$ but not uniformly. What is the relationship between the fourier series and the function $f$ in this case?