# fa.functional analysis – Decomposition of a function into right-sided and left-sided function

Here I define a distribution $$fin D’$$ to be right-sided if supp $$fsubseteq (0,infty)$$ and defnote it by $$f_+$$ and if the supp $$fsubseteq (-infty,0)$$ it is called left-sided and denoted by $$f_-$$.

Now, it is claimed that if $$f$$ is locally integrable function on $$mathbb{R}$$, then there is a unique decomposition $$f=f_++f_-$$ where $$f_+$$ is right-sided locally integrable function and $$f_-$$ is left sided locally integrable function.

For an example:

If I have $$A(omega)=frac{1}{omega^2+9}$$
then I can find a decomposition $$A_+(omega)=frac{i}{6(omega+3i)}$$ and $$A_-(omega)=frac{-i}{6(omega-3i)}$$ by inspection.

But, How do I find such a decomposition for function like
$$e^{-a x}theta(-x)$$ where $$theta$$ is Heaviside step function? Is there a general process to find such a decomposition?