# Definition:Integral Transform/Operator

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## 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