Definition:Integral Transform/Operator

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Definition

Let $F \left({p}\right)$ be an integral transform:

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


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.


Also denoted as

$F = T \left({f}\right)$ can be seen presented as $F = T f$ by some sources.


Sources