# Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation.

Is there a general theory of transformations?

My main interest is about classification of transformations satisfying specified properties like Parseval's identity or some inequalities.

Do you know a book or good website, where I can read a lot of interesting things about this topic( or is it a research topic)?

An integral transform is an operator that maps functions from one space to another. Formally $$T(f(x))=\int_{-\infty}^{\infty} K(x,y) f(y) dy$$
However, as much as it is fun to do work in the image space, one has to be able to interpret the results in the original space. To do so requires the study of the operator $K$. Usually one knows a priori the nature of the function $f$ by the nature of the problem one is dealing. Hence the study of integral transforms is the study of the operator $T$. Two properties come very easily $$T(f+g)=T(f)+T(g)\\ T(cf)=cT(f)$$