Definition:Inversion Theorem

Definition
Let $F \left({p}\right)$ be an integral transform:


 * $F \left({p}\right) = \displaystyle \int_a^b f \left({x}\right) K \left({p, x}\right) \, \mathrm d x$

Let $T$ be the integral operator associated with $F \left({p}\right)$:


 * $F = t \left({f}\right)$

An inversion theorem is a specification for an inverse integral operator $T^{-1}$ of the form $f = T^{-1} \left({F}\right)$ such that:
 * $f \left({x}\right) = \displaystyle \int_\alpha^\beta F \left({p}\right) H \left({x, p}\right) \, \mathrm d p$

should such an $H \left({x, p}\right)$ exist.

It is not necessarily the case that it does exist..