Definition:Integral Transform/Operator

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 {{{WRT|Integration}} $x$ 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.