I'd like to study some introductory properties of a specific type of integral transform, but I'm unsure where to look.
Let $f$ be an integral function such that $T[f](u) = \int_{t_1}^{t_2}f(t)K(t,s)$ for some suitable kernel $K$ such that the integral is well-defined.
How can I compute the inverse-kernel? How can I look at algebraic properties of sums and products and etc? Is there perhaps a chapter in a good book that gives an introduction to these types of transforms? What is the set of all kernels for which this is defined?
First of all, is $f$ being integrable a practical choice of study? Perhaps $f$ being integrable is not the best choice, perhaps I should instead say that $f$ is an element of $L_1$ space.
Is any such integral transform a map from $L_1$ to $L_1$? I just have lots of preliminary questions and I'm not sure what gives an introduction to these transforms specifically.
I have all these questions, but of course, authors writing research papers about real analysis often take the introductory information for granted and write exclusively for the nichest possible subset of readers, being those who have 30+ years of experience, rather than people with no experience, since they implicitly assume anyone interested in their paper would already be a prodigal master of analysis.
So probably, what I'm looking for is a specific chapter in some nice analysis book.