Let $M(\mathbb{T})$ be the space of (finite) complex measures on the circle ($[0,1]$ for calculation).
Let $BC(\mathbb{Z}) = l^\infty (\mathbb{Z})$ be the bounded continuous functions on the integers (bounded sequences).
Then the Fourier transform maps $\mathscr{F}: M(\mathbb{T}) \to BC(\mathbb{Z})$ by $\nu \mapsto \mathscr{F}(\nu) = (\widehat{f}(n) := \int_0^1 e^{-2 \pi i n x} d \nu (x))_{n=-\infty}^\infty$.
I know that $\mathscr{F}: L^1(\mathbb{T}) \to c_0(\mathbb{Z})$ has dense image (e.g. because the adjoint $\mathscr{F}^*$ is injective), but I don't know of a similar statement for $\mathscr{F}: M(\mathbb{T}) \to BC(\mathbb{Z})$
Follow-up: If the image is not dense, is there a better "upper bound" subspace of $BC(\mathbb{Z})$? Can we say anything more about the image of this map?