Is there a introductory, but also rigorous text in Integral transforms?

Question

I'm looking for a introductory textbook in integral transforms, especially Laplace and Fourier transforms.

While i'm looking for a introductory textbook, i also look for a textbook that have a good (not necessarily analysis) level of rigor. Do you know any text of this kind?

Thanks in advance.

Answer 1

There are various different points of view on the subject, and it is impossible to be sure which you're looking for given how little you've said.

The book by Dym & McKean on Fourier series and integrals contains a great variety of applications in geometry, physics, number theory, probability, and other things. It is somewhat rigorous. It has some chapters on material in analysis on which it relies, but that is better learned from other books. One oddity for which it can reasonable be criticized is the lack of use of generalized functions (Dirac's delta and its relatives), and I suspect that's because presenting the theory of those rigorously would have been too onerous. Many books go into the use of generalized functions in this topic without attempting to be rigorous at all.