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 $x$ 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 $\map f x$.