Definition:Integral Transform/Operator

Definition

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

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

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.

This article is complete as far as it goes, but it could do with expansion. In particular: The specific domain and codomain of $T$ need to be defined in order to make this rigorous. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also denoted as

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

Sources

• 1968: Peter D. Robinson: Fourier and Laplace Transforms ... (previous) ... (next): $\S 1.1$. The Idea of an Integral Transform