This is kind of a second part of this question: How do Integral Transforms work. There, I replied a comment asking about how can I find kernel for the inverse integral transform and then I was encouraged to create a new question.
In fact, it is about how Kernels work. For example, let's suppose I have created an Integral Transform whose Kernel is:
$$K(s,t) = \sqrt{s+t}$$ or $$K(s,t) = \ln(s+t)$$
How can I find/calculate its inverse? Or more generally: given a Kernel, how should I proceed to get its kernel for the inverse integral transform?
Also, what are the possibilities of creating a Kernel, i.e. what should $K$ satisfy? Additionally, what are the advantages of Laplace's Kernel ($e^{-st}$)?
Which books, papers or anything would you recommend me to read? I only know Calculus (Single-Variable, Multi-Variable and Vectorial), a bit of ODEs and some Linear Algebra. Thanks