The Wiener algebra $W:=W(\mathbb{T}^n)$ on the torus is defined as the algebra of all continuous fonctions $f$ on $\mathbb{T}^n$ such that $(\widehat f(k))\_{k\in \mathbb{Z}^n} \in \ell^1(\mathbb{Z}^n)$. This is equivalent to say that 
the family $(\widehat f(k) e\_k)\_{k\in \mathbb{Z}^n}$ (where $(e\_k)$ is the Fourier basis) is absolutely summable in the Banach space $C(\mathbb{T}^n)$. 


Define $W'$ the set (containing $W$) of all integrable functions $f$ on the torus such that $(\widehat f(k) e\_k)\_{k\in \mathbb{Z}^n}$ is summable (but not necessarily absolutely) in $C(\mathbb{T}^n)$.


Question : Do we have $W=W'$ ? If not, has this space $W'$ been explored?



On a weaker condition of summability for Fourier series
If a series $\sum c(k) e\_k$ is summable in $C({\mathbb T}^n)$, in particular, the series $\sum c(k) e\_k(0)=\sum c(k)$ is summable (in $\mathbb{C}$), thus $\sum |c(k)|<\infty$. 


