What is the abstract definition of a Mathematical transform?

Question

The question is What is the abstract definition of a Mathematical transform? Is there any book talking specifically about what makes a Mathematical transform a Mathematical transform?

I am particularly talking about Fourier, Laplace, Wavelet and other transforms of the same kind. For instance, how can I know the map I have devised is a transform or not?

Furthermore, any books and references regarding the question would be much appreciated.

Answer 1

What is the abstract definition of a Mathematical transform? ... I am particularly talking about Fourier, Laplace, Wavelet and other transforms of the same kind.

All three of those are "integral transforms", defined in places like the English Wikipedia page for "integral transform" as:

any transform $T$ of the ... form: $(Tf)(u)={\displaystyle \int_{t_1}^{t_2}}f(t)K(t,u)\,\mathrm dt$

---

For instance, how can I know the map I have devised is a transform or not?

If you defined it via an integral like the above, it's an integral transform. If you didn't, then it isn't an integral transform, but there's a very tiny chance it coincidentally equals an integral transform.

---

Furthermore, any books and references regarding the question would be much appreciated.

This is not my field of study, but I imagine books with "integral transforms" in their title might be appropriate or help you find other sources, depending on your interests/goals. For example, "Integral Transforms and their Applications", "Introduction to Integral Transforms", "Integral Transforms and Operational Calculus", "Integral Transforms in Applied Mathematics", etc.

If you have a more specific question about the best book for a certain goal, that might be worth making a separate question on Math StackExchange with the full context.