# Definition:Integral Transform

## Definition

Let $p$ be a variable whose domain is a subset of the set of real numbers $\R$.

Let $\closedint a b$ be a closed real interval for some $a, b \in \R: a \le b$.

Let $f: \closedint a b \to \R$ be a real function defined on the domain $\closedint a b$.

Let $\map K {p, x}$ be a real-valued function defined for all $p$ in its domain and all $x \in \closedint a b$.

Let $\map f x \map K {p, x}$ be integrable with respect to $x$ for all $p$ in its domain and all $x \in \closedint a b$.

Consider the real function $\map F p$ defined as:

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

Then $\map F p$ is an **integral transform** of $\map f x$.

### Kernel

The function $\map K {p, x}$ is the **kernel** of $\map F p$.

### Integral Transform as Integral Operator

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.

### Image Space

The domain of $p$ is known as the **image space** of $T$.

## Sources

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