How is the word kernel associated with distributions?

Question

I am trying to rationalize the meaning of the term kernel, especially when it is associated with distributions. The English and German etymology all show that the literal meaning is corn (English) and grain (in German). For example the term Gaussian kernel is often used in digital filtering to remove noise from signal. According to the Earliest Known Uses of Math Words website kernel entered statistics with the use of Fourier theory in describing estimates for spectral density and probability density functions. However it does not explain the choice.

What is the rationalization behind the choice of this word, I mean what is common in the original meaning of kernel and window functions as shown on Wikipedia? The term window function makes much more sense.

Answer 1

Rationalizations that "make sense" are often urban legends after the fact, people who introduce terms rarely make a point of it or report their reasons. The process of spreading is largely by accident and loose association, sometimes aided by invented backstories.

Fredholm used the French version ("noyau") for integral equations, for whatever reasons, perhaps because the kernel is the "core" of the integral. It stays while the unknown function varies. Hilbert and Bochner took after him. It is similar to the integral "core" for Fourier and other sums, as well as to integral representations of solutions to boundary value problems. So Zygmund's monograph extended the use. With big names backing it, it spread. When statisticians started using Fourier series, they picked it up from Bochner, Zygmund and others.

"Gaussian kernel" refers to taking convolutions with the Gaussian distribution, which are similar (identical in the simplest case) to convolutions with the heat kernel to solve the heat equation, or convolutions with the Dirichlet/Fejér kernels to approximate Fourier series, or, indeed, to forming an integral equation. So this association makes more sense than many others.

The word is also used in algebra, coming from different people and for unrelated reasons.