$\DeclareMathOperator\supp{supp}$Let $H:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ be the Hilbert transform. Let suppose we have a compaclty supported function $f \in L^{2}(\mathbb{R})$ such that $\supp(f)\subseteq \mathbb{R}_{+}$, the support of $H(f)$ is probably no longer in $\mathbb{R}_{+},$ on the other hand if we have an arbitrary $g \in L^{2}(\mathbb{R})$ is well know that $g=H(-Hg)$, that is $g$ is the Hilbert transform of someone.

Again, given an arbitrary $g \in L^{2}(\mathbb{R})$, is it possible to find a $f \in L^{2}(\mathbb{R})$ such that $\supp(f)\subseteq \mathbb{R}_{+}$ and $Hf$ is near $g$ ? Or putit in another form is the set $\{H(f)/ f \in L^{2}(\mathbb{R}) \wedge \supp(f)\subseteq \mathbb{R}_{+}\}$ dense in $L^{2}(\mathbb{R})$?

Another related question, given $g \in L^{2}(\mathbb{R})$ such that $\supp(g)\subseteq \mathbb{R}_{-},$ is it possible to find a $f \in L^{2}(\mathbb{R})$ such that $\supp(f)\subseteq \mathbb{R}_{+}$ and $\chi_{\mathbb{R}_{-}}(x)Hf(x)=g(x)$? In this last question I want $Hf$ equals $g$ only over the negatives, so any $f \in L^{2}(\mathbb{R})$ such that $\supp(f)\subseteq \mathbb{R}_{+}$ who satisfies $Hf=\hat{g}$ (where $\hat{g}$ and $g$ coincide over the negatives) will be useful.