Is there an $f\in L^{1}(T)$ whose partial sums of Fourier series $S\_{n}(f)$ satisfies $\|S\_{n}(f)\|\_{L^{1}(T)} \rightarrow \|f\|\_{L^{1}(T)}$ but $S\_{n}(f)$ fails to converge to $f$ in $L^1$-norm ?



On the Existence of Certain Fourier Series
I was hesitating for a while whether to answer or to vote to close and to refer the OP to AoPS, but, since the question has been upvoted, here goes.


Suppose that $f\_k\in L^1$ converges to $f\in L^1$ in the sense of distributions and in measure. Suppose also that $\|f\_k\|\_1\to \|f\|\_1$. Then $f\_k\to f$ in $L^1$. 


Indeed, let $g$ be a bounded by $1$ infinitely smooth function such that $\int fg>\|f\|\_1-\delta$. Then $\int f\_kg > \|f\|\_1-2\delta$ for large $k$. Now, $\int\_{\{|f\_k-f|<\delta\}} f\_kg\ge \int\_{\{|f\_k-f|<\delta\}} fg-\delta\ge \int fg-2\delta\ge \|f\|\_1-3\delta$ for large $k$ because the integral of a fixed $L^1$ function $fg$ over a set of small measure is small. 


So, $\int|f\_k|\ge \int\_{\{|f\_k-f|\ge\delta\}} |f\_k|+\|f\|\_1-3\delta$ whence the first integral is at most $4\delta$ for large $k$. Thus
$$
\int |f\_k-k|\le \int\_{\{|f\_k-f|\ge \delta\}} (|f\_k|+|f|)+\delta\le 6\delta
$$
for large $k$.


To apply this to $f\_k=S\_kf$, one only needs to check the convergence in measure. But it immediately follows from the weak type 1-1 bound for $S\_k$ (applied to the difference of $f$ and a trigonometric polynomial approximating $f$ in $L^1$, of course).


