Category:Definitions/Integral Transforms

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This category contains definitions related to Integral Transforms.
Related results can be found in Category:Integral Transforms.

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

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval for some $a, b \in \R: a \le b$.

Let $f: \left[{a \,.\,.\, b}\right] \to \R$ be a real function defined on the domain $\left[{a \,.\,.\, b}\right]$.

Let $K \left({p, x}\right)$ be a real-valued function defined for all $p$ in its domain and all $x \in \left[{a \,.\,.\, b}\right]$.

Let $f \left({x}\right) K \left({p, x}\right)$ be integrable with respect to $x$ for all $p$ in its domain and all $x \in \left[{a \,.\,.\, b}\right]$.

Consider the real function $F \left({p}\right)$ defined as:

$F \left({p}\right) = \displaystyle \int_a^b f \left({x}\right) K \left({p, x}\right) \, \mathrm d x$

Then $F \left({p}\right)$ is an integral transform of $f \left({x}\right)$.