# Definition:Inversion Theorem

## Definition

Let $\map F p$ be an integral transform:

$\map F p = \ds \int_a^b \map f x \map K {p, x} \rd x$

Let $T$ be the integral operator associated with $\map F p$:

$F = \map t F$

An inversion theorem is a specification for an inverse integral operator $T^{-1}$ of the form $f = \map {T^{-1} } F$ such that:

$\map f x = \ds \int_\alpha^\beta \map F p \map H {x, p} \rd p$

should such an $\map H {x, p}$ exist.

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