First of all, I think the description of the problem is a little bit problematic since
it does not define $\theta(0)$. I would use the convention that $\theta(0)=0$
Here is a reference:
https://en.wikipedia.org/wiki/Heaviside_step_functionFor
$${f_ + }(x) = {e^{ - \varepsilon |x|}}\theta ( + x) = \left\{ {\matrix{
{{e^{ - \varepsilon x}}} & {x \ge 0} \cr
0 & {x < 0} \cr
} } \right.$$
$${{\hat f}_ + }(k) = {1 \over {2\pi }}\int_0^\infty {{e^{ - (\varepsilon + ik)x}}dx = } \left. { - {{{e^{ - (\varepsilon + ik)x}}} \over {2\pi (\varepsilon + ik)}}} \right|_0^\infty = {1 \over {2\pi (\varepsilon + ik)}}$$
Similarly,
$${{\hat f}_ - }(k) = {1 \over {2\pi }}\int_{ - \infty }^0 {{e^{(\varepsilon - ik)x}}dx = } \left. {{{{e^{(\varepsilon - ik)x}}} \over {2\pi (\varepsilon - ik)}}} \right|_{ - \infty }^0 = {1 \over {2\pi (\varepsilon - ik)}}$$
Therefore,
$${f_ + }(x) = \int_{ - \infty }^\infty {{1 \over {2\pi (\varepsilon + ik)}}} {e^{ikx}}dk$$
$${f_ - }(x) = \int_{ - \infty }^\infty {{1 \over {2\pi (\varepsilon - ik)}}} {e^{ikx}}dk$$
And
$$\hat g(x) = {{\hat f}_ + }(k) + {{\hat f}_ - }(k) = {1 \over {2\pi (\varepsilon + ik)}} + {1 \over {2\pi (\varepsilon - ik)}} = {{2\varepsilon } \over {2\pi ({\varepsilon ^2} + {k^2})}}$$
$$\hat h(x) = {{\hat f}_ + }(k) - {{\hat f}_ - }(k) = {1 \over {2\pi (\varepsilon + ik)}} - {1 \over {2\pi (\varepsilon - ik)}} = {{ - 2ik} \over {2\pi ({\varepsilon ^2} + {k^2})}}$$
