Riesz Representation Theorem for $L^2(mathbb{R}) oplus L^2(mathbb{T})$?

The spaces $L^2(mathbb{R})$ (square-integrable functions) and $L^2(mathbb{T})$ (1-periodic square-integrable functions, considered over the real line $mathbb{R}$) are two subspaces of the space of tempered distributions $mathcal{S}'(mathbb{R})$ and one can easily show that the sum $L^2(mathbb{R}) oplus L^2(mathbb{T})$ is direct.

The duals of $L^2(mathbb{R})$ and $L^2(mathbb{T})$ are isometrically isomorphic to $L^2(mathbb{R})$ and $L^2(mathbb{T})$, respectively (Riesz representation theorem). Therefore, the continuous dual of the direct sum is simply $L^2(mathbb{R}) oplus L^2(mathbb{T})$ in the sense that (1) an element $g_1 + g_2 in L^2(mathbb{R}) oplus L^2(mathbb{T})$ defines a continuous linear functional over $L^2(mathbb{R}) oplus L^2(mathbb{T})$ via
$$(f_1 + f_2) mapsto langle f_1 , g_1 rangle_{L^2(mathbb{R})} + langle f_2 , g_2 rangle_{L^2(mathbb{T})}$$
(which uses that both decompositions $f = f_1 + f_2$ and $g = g_1+g_2$ are unique), and that (2) any element of $(L^2(mathbb{R}) oplus L^2(mathbb{T}))’)$ is of this form.

I would like to identify the subset $mathcal{X}subset mathcal{S}'(mathbb{R})$ of functions $g$ such that
$$ L^2(mathbb{R}) oplus L^2(mathbb{T}) ni f_1 + f_2 mapsto int_{mathbb{R}} g(x) (f_1 + f_2)(x)mathrm{d}x$$
specifies a continuous linear functional over $L^2(mathbb{R}) oplus L^2(mathbb{T})$.
Clearly, by restricting it to $L^2(mathbb{R})$ (i.e. setting $f_2=0$), we need to have $g in L^2(mathbb{R})$. Moreover, $mathcal{X}$ contains any square-integrable compactly supported functions, but also functions that are not compactly supported but that have sufficiently nice asymptotic properties such that the integral $int_{mathbb{R}} g (x) f_2(x)mathrm{d}x$ is well-defined for any square-integrable periodic $f_2$ and defines a continuous functional over $L^2(mathbb{T})$.

Question: Is there a way to identify the space $mathcal{X}$ I am depicting? Can we reach any linear functionals over $L^2(mathbb{R}) oplus L^2(mathbb{T})$ by doing so? I am also interested by generalization to other direct sums between spaces of periodic and non-periodic functions (e.g., $L^p$-spaces, or spaces of continuous-functions for the supremum norm).