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?