Definition:Integral Transform/Operator

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Definition

Let $\map F p$ be an integral transform:

$\map F p = \displaystyle \int_a^b \map f x \map K {p, x} \rd x$


This can be written in the form:

$F = \map T f$

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

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


Thus $T$ transforms the function $\map f x$ into its image $\map F p$, which is itself another real function.


Also denoted as

$F = \map T f$ can be seen presented as $F = T f$ by some sources.


Sources