Integral Transformation
Definition
If a map $J$ from a function space to a function space is defined as the following integral, then $J$ is called an integral transform.
$$ (Jf) (x) = \int_{a}^{b} K(x,t)f(t)dt $$
$$ J : f(\cdot) \mapsto \int_{a}^{b} K(\cdot,t)f(t)dt $$
In this case, $K$ is referred to as the kernel of $J$. If a map from $Jf$ to $f$ exists, it is denoted as $J^{-1}$ and called the inverse transform of $J$.
Description
Since integration is linear, integral transforms are linear transforms.
The integration domain does not necessarily have to be bounded. It can be $a=-\infty$, $b=\infty$, or both. Although integral transforms can be created arbitrarily according to the definition, to have meaningful interpretations, solving the given problem should be easier in terms of $Jf$ than in $f$, or an inverse transformation should exist, allowing free conversion between $Jf$ and $f$. Examples of integral transforms include the following.
Fourier transform $\mathcal{F}$:
$$ \mathcal{F}f(\xi)=\int _{-\infty} ^{\infty} f(x)e^{i \xi x}dx,\quad K(x,\xi)=e^{i\xi x} $$
Laplace transform $\mathcal{L}$:
$$ \mathcal{L}f(s)=\int _{0} ^{\infty}f(t)e^{-st}dt,\quad K(t,s)=e^{-st} $$
Mellin transform $\mathcal{M}$:
$$ \mathcal{M}f(s)=\int_{0}^{\infty} f(x)x^{s-1}dx,\quad K(x,s)=x^{s-1} $$
Radon transform $\mathcal{R}$:
$$ \mathcal{R}f(s,\theta)=\int_{-\infty}^{\infty}f(s\cos\theta-t\sin\theta, s\sin\theta+t\cos\theta)dt $$