Why are these functions called "kernels"?

Question

In the last years while studying numerical analysis I came across different "kernels", like the Dirichlet Kernel

$$D_n(x) = \sum_{k=-n}^n e^{ikx}$$

the Fejer-Kernel

$$F_n(x) = \frac{1}{n} \sum_{k=0}^{n-1} D_k(x)$$

or the Peano-Kernel

$$K_m(t) := \frac{1}{m!}R_x[(x-t)_{+}^m].$$

I don't understand why these functions are called "kernels" though. They seem to have nothing to do with the sort of kernel we know from linear algebra. Do these functions have something special in common? Or are there historical reasons?

Answer 1

The usage of the word "kernel" in that context and in linear algebra appear to be coincidental.

The use of kernel in algebra appears to be unrelated to its use in integral equations and Fourier analysis. The OED gives the following quotation from Pontrjagin's Topological Groups i. 11 (translated by E. Lehmer 1946) "The set of all the elements of the group G which go into the identity of the group G* under the homomorphism g is called the kernel of this homomorphism."

(Quoted from Earliest Known Uses of Some of the Words of Mathematics (K))

There one also finds further information on the origin of the word kernel in your context. I quote a part.

Ivar Fredholm used the French word, “noyau,” in his famous paper on INTEGRAL EQUATIONS, “Sur une classe des équations fonctionnelles,” Acta Math., 27, (1903), 365–390. David Hilbert put this into German as Kern [...] The English word kernel appears in M. Bôcher’s Introduction to the Study of Integral Equations (1909): “K is called the kernel of these equations.”

I just looked up the Hilber paper; he says nothing why that word is used. I'd speculate it is simply because it is somehow inside the integral when one does a convolution. (I will try to look up the Fredholm paper too.)