The Wiener algebra $A(\mathbb T)$ is defined as the space of all functions defined on $\mathbb T$(the torus) such that its fourier series satisfies: $$\|f_n\|_{A(\mathbb T)}:=\sum\limits_{n}|\hat f(n)|<\infty.$$
An immediate consequence is stated: let $f_n\in A(\mathbb T)$ and $\|f_n\|\leq 1.$ Assume that $f_n$ converge to $f$ uniformly on $\mathbb T$. Then $f\in A(\mathbb T)$ and $\|f\|\leq 1.$
Here is my proof which can be skipped for readers who are not interested in:
Proof: Since $f_n$ converges uniformly to $f$, then for any $m>0$, there exists $N_m$ such that if $n\geq N_m$, $$\sup_t|f(t)-f_n(t)|<\frac{1}{m}.$$ For any $m\in\mathbb Z, n\geq N_m$, $$|\hat f(k)-\hat f_n(k)|=\frac{1}{2\pi}\int_{\mathbb T}|f(t)-f_{n}(t)|e^{-ikt}dt\leq\frac{C}{mk},$$ for any $k\in\mathbb Z.$
Thus it is obvious that for $n\geq N_m$, there holds $$\sum\limits_{|k|\leq m}|\hat f|=\sum\limits_{|k|\leq m}(|\hat f(k)-\hat f_n(k)|+|\hat f_n(k)|)\leq 2\sum\limits_{j=1}^m\frac{1}{mj}+1\leq 2\frac{\log m}{m}+1.$$ As $m\rightarrow \infty,\ \|f\|_{A(\mathbb T)}\leq 1.$
Q.E.DIs there any problems in it? I can't find any errors in it.
My central problem is described as follows:
Problem: the conditions do not imply $\lim\|f-f_n\|_{A(\mathbb T)}=0$, i believe this conclusion is correct, however, i cannot find some counterexamples. Can someone tell me the counterexample: my idea is to construct $$ \widehat{f_n}(m)=\left\{ \begin{aligned} &\frac{\mathrm{sgn}(m)}{n}& m\in[-n,n]\backslash\{0\}\\ &0 & else. \end{aligned} \right. $$However, i can not find it uniformly converges to $0.$
This problem follows from Yitzhak Katznelson "An introduction to harmonic analysis."
