Definition talk:Integral Transform/Operator

Thoughts on incorporating this
I am looking for a way to define the integral operator as follows:

Let $\CC \closedint a b$ be the space of real-valued functions continuous on closed interval.

Let $x \in \CC \closedint a b$

Let $\R$ be the set of real numbers.

The integral operator, denoted by $I$, is the mapping $I : \CC \closedint a b \to \R$ such that:


 * $\ds \map I x := \int_a^b \map x t \rd t$

In some sense this has a trivial kernel, i.e. 1. However, the point of this is to emphasize that the input, or the variable, is $\map x t$. Can we include $\map x t$ or $\map f x$ as one of variables of an integral transform? It may happen that I will later need to consider continuity or differentiation wrt $f$ or $x$.

In other words, I need to replace $\map F p = \displaystyle \int_a^b \map f x \map K {p, x} \rd x$ with something like :$\map F {f, p} = \displaystyle \int_a^b \map f x \map K {p, x} \rd x$. Any thoughts?--Julius (talk) 13:31, 21 February 2021 (UTC)