Definition:Integral Transform

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Definition

Let $p$ be a variable whose domain is a subset of the set of real numbers $\R$.

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval for some $a, b \in \R: a \le b$.

Let $f: \left[{a \,.\,.\, b}\right] \to \R$ be a real function defined on the domain $\left[{a \,.\,.\, b}\right]$.

Let $K \left({p, x}\right)$ be a real-valued function defined for all $p$ in its domain and all $x \in \left[{a \,.\,.\, b}\right]$.

Let $f \left({x}\right) K \left({p, x}\right)$ be integrable with respect to $x$ for all $p$ in its domain and all $x \in \left[{a \,.\,.\, b}\right]$.


Consider the real function $F \left({p}\right)$ defined as:

$F \left({p}\right) = \displaystyle \int_a^b f \left({x}\right) K \left({p, x}\right) \, \mathrm d x$


Then $F \left({p}\right)$ is an integral transform of $f \left({x}\right)$.


Kernel

The function $K \left({p, x}\right)$ is the kernel of $F \left({p}\right)$.


Integral Operator

This can be written in the form:

$F = T \left({f}\right)$

where $T$ is interpreted as the (unitary) operator meaning:

Multiply this by $K \left({p, x}\right)$ and integrate {with respect to $x$ between the limits $a$ and $b$.


Thus $T$ transforms the function $f \left({x}\right)$ into its image $F \left({p}\right)$, which is itself another real function.


Image Space

The domain of $p$ is known as the image space of $T$.


Sources