Let $\mathbb{T}$ represent the $[0,1)$ interval and for each $w\in\mathbb{Z}$ denote the function $$e_w:\mathbb{T}\to\mathbb{C}$$ such that $$\forall x\in\mathbb{T}, \ e_w(x)=e^{2\pi iwx}.$$ It is known that the set of harmonic oscillators $e_w$ is a complete system in $L_p(\mathbb{T})$, i.e. such spaces are "generated" by $\big\{ e_w : \ w\in\mathbb{Z} \big\}$ with coefficients in some specific set of series $\ell^q\subset\big\{(c_w)_{w\in\mathbb{N}}: \ c_w\in\mathbb{C}\big\}$ since the set of finite sums $\big\{ \sum\limits_{w=0}^{k} c_we_w\big\}$ is dense in it, that is
$\forall f \in L_p(\mathbb{T}), \ \exists (c_w)_{w\in\mathbb{N}}\in \ell^q \ \Big( \lim\limits_{k\to \infty} \sum\limits_{w=0}^{k} c_w e_w = \ f \Big)$.
For each $p\in (0,\infty], \ (c_w)_{w\in\mathbb{N}}$ lies in a specific set of complex series $\ell^q$ where $1=\dfrac{1}{p} + \dfrac{1}{q}$.
My question is: given the set of all complex sequences $\{ (c_w)_{w\in\mathbb{N}} : \ c_w\in\mathbb{C}\}$, What is the space generated by the topological closure of the finite sums $\sum\limits_{w=0}^{k} c_w e_w$ ? In other words, What is $$\overline{\bigg\{ \sum\limits_{w=-n}^{n} c_w e_w : \ n\in\mathbb{N}, \ c_w\in\mathbb{C}\bigg\} } \ ?$$
Evidently, several of these funcions reach infinite values in various subsets.