For any invertible integral transform $T$ of kernel $K$ that maps a function $f$ to the function $\varphi$ such that $$\varphi(s)=\left[T\left\lbrace f\right\rbrace\right](s)=\int_a^bK(x,s)f(x)dx$$
How can we show that their is at most only one possible inverse Kernel $K^{-1}$ ?
For each of the usual integral transforms (the Fourier transform, the Laplace transform, and the Mellin transform), their kernel is said to be invertible because their respective inverse transforms can all be expressed as integral transforms with another kernel, called the inverse kernel of $K$.
Now, how do we know that for each of these transform, the inverse kernel is unique ?
Naively, I can imagine a scenario where, although the inverse transform is known to exist, we don't have a analytical expression of it valid on all of its domain, but rather two integral transforms with each a different Kernel, defined on two disjoint subset of the domain of the inverse transform, such that the inverse transform coincides with each of these on their respective domains...
Should an invertible integral transform check particular requirements in order to ensure the uniqueness of its inverse kernel ?
Any insight ?
