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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)?

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An integral transform is an operator that maps functions from one space to another. Formally $$ T(f(x))=\int_{-\infty}^{\infty} K(x,y) f(y) dy $$

Now the practical motivation for an integral transform is to reduce the complexity of the problem i.e the the mathematical operations will be much easier to handle in the image space.

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 $$ T(f+g)=T(f)+T(g)\\ T(cf)=cT(f) $$

To ensure invertibility, one has to show that the kernel space only contains the null fucntion.

The abstract study of integral transforms is Fredholm's theory. Refer to integral transforms and Applications by Debnath to get a detailed information on the different kinds of transforms.

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Have a look at this reference by Ahmed Zayed

Handbook of Function and Generalized Function Transformations.

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