I have a function $f(u)$ satisfying the following properties $$ \lim_{u\to\pm\infty} f(u) = f^\pm,~~ \lim_{u\to\pm\infty} f'(u) \sim {\cal O} \left( |u|^{-3/2} \right) = 0 $$ The function $f(u)$ can be written as $$ f(u) = i \int_0^\infty d\omega \left( g_1(\omega)e^{-i \omega u} + g_2(\omega)e^{i \omega u} \right) $$ My goal is now the following - What are $g_1(\omega)$ and $g_2(\omega)$ near $\omega = 0$ in terms of $f^\pm$.
Let me get started as an example of what I'm looking for. We have $$ \int_{-\infty}^\infty f'(u) du = \int_{-\infty}^\infty du \int_0^\infty d\omega~ \omega \left( g_1(\omega)e^{-i \omega u} - g_2(\omega)e^{i \omega u} \right) = \frac{1}{2} \lim_{\omega \to 0} \left[ \omega \left( g_1(\omega) - g_2(\omega) \right) \right] $$ The LHS is simply $f^+ - f^-$. Thus $$ f^+ - f^- = \frac{1}{2} \lim_{\omega \to 0} \left[ \omega \left( g_1(\omega) - g_2(\omega) \right) \right] $$ I'm looking for similar expressions but one involving only $g_1(\omega)$ or $g_2(\omega)$. Another thing I tried was $$ \int_{-\infty}^\infty du~f(u) = \frac{i}{2} \lim_{\omega \to 0} \left[ g_1(\omega) + g_2(\omega) \right] $$ However, I now do not know how to relate the LHS to $f^+$ and $f^-$.
Any ideas?
Please tell me if this question is vague, I will try to clarify. Thanks!