Intuition about Dirichlet Kernel

Question

I am studying Fourier analysis from the text "Stein and Shakarchi" and there is this thing on Dirichlet Kernel. It's fine to define it as a trigonometric poylnomial of degree $n$ , but what is the mathematical intuition behind calling it a Kernel ? I have also thought of Kernel as being a set of zeroes of sum function. Is there a relation between both the terminologies?

Answer 1

In general, if you have a linear operator $T$ on some space of functions, defined by an integral $$Tf(x)=\int f(t) K(x,t)\,dt,$$then $K$ is the "kernel".

Answer 2

The "kernel" of something is the essential part of it, the germ of it, a whole seed. That's a definition that makes sense in terms of the linear operator $T$, because the essential part of it is the function $K$. To me, it makes no sense how the null space came to be called "kernel." I would guess that the use of kernel in terms of the integral operator came before the abstract definition of the null space of a linear operator; integral operators were around before general linear operators.