# How to find the inverse of an integral transform?

• I
I'm trying to find the distribution of a random variable ##T## supported on ##[t_1, t_2]## subject to ## \mathbb{E}[V(t', T)] = K, \forall t' \in [t_1, t_2]##. In integral form, this is : $$\int_{t_1}^{t_2} V(t', t).f(t) \, dt = K,\forall t' \in [t_1, t_2],$$ which is just an exotic integral transform.

So if I can find the inverse transform, I'm done. But the function ##V## is ...not nice. It's nowhere differentiable for ##t < t'##, there's a jump at ##t = t'## and it's constant from then on.

So do you guys know of any methods to find the inverse to an integral transform with a nasty kernel like ##V##? Or can you see another way to solve these equations? I'd even be happy with numerical techniques.