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