Denote Mon$_\infty \subseteq \mathbb{C}[z_1,z_2,...]$ the set of monomials on a countable number of variables, that is, elements of the form $$ z_{i_1}^{k_1}z_{i_2}^{k_2}...z_{i_n}^{k_n}, $$ where each $i_j,k_j \in \mathbb N$.
My question is if there is an injective map Mon$_\infty \rightarrow C(S^1)$, that to each monomial assigns a continuous function on the unit circle $S^1 \subseteq \mathbb C$.
Im thinking things analogous to Fourier Transform or other integral/sum transforms.
If necessary, the target space can be enlarged to $L^2(S^1)$.