Question

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 ?

Answer 1

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).