Integral Transform

Topics: Integral Transform

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An integral transform is an operator that is derived from other functions though a definite integral. They allow us to switch from a reference system to another, with the advantage of being able to go back to the original system (this ability being the inverse transform).

Instances of integral transforms include the Laplace transform.

(definition)

Formally, an integral transform is an operator that associates a new function to a given set through integration with respect to a given parameter:

$$T\{f(t)\}=F(s)={\int }_a^{b}K(t,s)f(t)dt$$

…where:

• $K(t,s)$ is the integral kernel of the transform
• $(a,b)$ is the interval that is commonly infinite
• $F(s)$ is the transform of the function $f$

Do note that integral transforms are linear transformations.