This covers all transformations of functions by integrals, including but not limited to the Radon, X-Ray, Hilbert, Mellin transforms. Use (wavelet-transform), (laplace-transform) for those respective transforms, and use (fourier-analysis) for questions about the Fourier and Fourier-sine/cosine transforms.

Integral transformations have been successfully used for almost two centuries in solving many problems in applied mathematics, mathematical physics, and engineering science. Historically, the origin of the integral transforms including the Laplace and Fourier transforms can be traced back to celebrated work of P. S. Laplace (1749–1827) on probability theory in the 1780s and to monumental treatise of Joseph Fourier (1768–1830) on La Théorie Analytique de la Chaleur published in 1822.

The integral transform of a function $~f(x)~$ defined in $~a ≤ x ≤ b~$ is denoted by $~\mathcal I \{f(x)\} = F(p)~$, and defined by $$~\mathcal I \{f(x)\} = F(p)~=\int_a^bf(x)~K(x,p)~dx$$where $~K(x,t)~$ is called the integral kernel of the transform. The operator $~\mathcal I~$ is usually called an

integral transform operatoror simply anintegral transformation. The transform function $~F(p)~$ is often referred to as the image of the given object function $~f(x)~$ , and $~p~$ is called the transform variable.

Similarly, the integral transform of a **function of several variables** is defined
by $$~\mathcal I \{f(x)\} = F(p)~=\int_Sf(x)~K(x,p)~dx$$where $~x=(x_1,~\cdots~,~x_n)~$,$~~p=(p_1,~\cdots~,p_n)~$, and $~S ⊂ \mathbb R^n~$.

A mathematical theory of transformations of this type can be developed by using the properties of Banach spaces. From a mathematical point of view, such a program would be of great interest, but it may not be useful for practical applications.

**References:**

https://en.wikipedia.org/wiki/Integral_transform

"Integral Transforms and Their Applications" by Lokenath Debnath, Dambaru Bhatta