Why is the kernel of an integral transform called kernel?

Question

I mean, in mathematics things with the same name are usually related. So, what is the relationship between the kernel of an integral transform and the kernel of an linear transformation? If it is none, why the kernel of an integral transform is called like that?

There is another post here that says that there is no relationship but I don't think that is true.

Answer 1

The kernel of an integral transform is called kernel with more right than the kernel (nullspace) of a linear transformation. The two meanings have absolutely nothing to do with each other.

An integral transform is a black box that takes some function $f$ as input and produces some (other) function $Tf$ as output. The kernel is what's inside that box. It is the "program" that encodes the exact manipulations to be applied to $f$.

Answer 2

If you think about it,

a) the kernel completely determines the integral transform (hence is the "core"

b) the transforms look like $\int (-) K dx$ (hence $K$ is the "core")