It is a standard fact of harmonic analysis that the span of the functions $$g_z(t) = \mbox{Im } (t-z)^{-1},$$ ranging over all $z \in \mathbb{C}$ with $\mbox{Im } z > 0$, is dense in $C_0(\mathbb{R})$ with respect to the uniform norm, and this implies density in a variety of other spaces as well. Is there a known characterization of the closure in a "standard function space" of the span of $$\{g_z(t) : \mbox{Im } z > \epsilon\}$$ for some fixed $\epsilon > 0$?
I'm keeping "standard function space" intentionally vague so as not to disqualify a potential satisfactory answer in $L^p$ by insisting on the space being $C_0$, etc.
Thanks in advance.
Let $\mathcal{M}_{p}$ be the closure of $\{ \Im(t-z)^{-1} : \Im z > \epsilon > 0 \}$ in $L^{p}$ for some $1 \le p < \infty$, and suppose $\mathcal{M}_{p} \ne L^{p}(\mathbb{R})$. Then there exists $f \in L^{q}(\mathbb{R}) = (L^{p})^{\star}$ such that $$ F(z)= \frac{1}{\pi}\int_{-\infty}^{\infty}f(t)\Im\frac{1}{t-z}dt =0,\;\;\;\Im z > \epsilon > 0. $$ That gives you a harmonic function $F$ in the upper half plane that is identically $0$ for $\Im z > \epsilon > 0$. Hence $F$ must be identically $0$ in $\Im z > 0$. So, $f=0$ a.e., which implies that $\mathcal{M}_{p}=L^{p}$ for $1 \le p < \infty$. You can view $C_0$ as a subspace of $C(\mathbb{T})$, where $\mathbb{T}$ is the 1-d torus, and the dual gives you a complex Borel measure on $\mathbb{R}$ for which $$ M(z) = \frac{1}{\pi}\int_{\infty}^{\infty}\Im\left( \frac{1}{t-z}\right)d\mu(t) = 0,\;\;\; \Im z > \epsilon > 0. $$ The same conclusion applies: $M$ must be identically $0$ for $\Im z > 0$. That's enough to imply that $\mu I=0$ for every finite interval $I$ because $$ \frac{1}{2}\{\mu[a,b]+\mu(a,b)\} = \lim_{\epsilon\downarrow 0}\int_{a}^{b}F(x+i\epsilon)dx. $$