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Tf is a function of a parameter, denoted u in the equation above. Thus, an integral transform maps one function into another which is a function of the parameter.
There are several useful integral transforms. Each transform corresponds to a different choice of the function g, which is called the kernel of the transform.
| Transform | Symbol | Kernel |
|---|---|---|
| Laplace transform | ||
| Fourier transform | ||
| Hilbert transform | ||
| Mellin transform | ||
| Identity transform |
Although the properties of integral transforms vary widely, they have some properties in common. For example, every integral transform is a linear operator, since the integral is a linear operator, and in fact if the kernel is allowed to be a generalized function then all linear operators are integral transforms (a properly formulated version of this statement is the Schwartz kernel theorem ).