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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?

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  • $\begingroup$ They are used as integrand multipliers, if i am not wrong. See kernel function $\endgroup$
    – mvw
    Commented May 13, 2015 at 23:36

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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.)

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  • $\begingroup$ I could not easily locate it in that paper for now. $\endgroup$
    – quid
    Commented May 14, 2015 at 0:02
  • $\begingroup$ Volterra came before Fredholm, with Volterra integral equations a special case of Fredholm integral equations. And Abel came long before Volterra with a special case, Abel's integral equation. Did either Abel or Volterra, or someone else writing about their work, use the term "kernel" before Fredholm? $\endgroup$ Commented May 14, 2015 at 0:16
  • $\begingroup$ If the source I quote is to be trusted: no. $\endgroup$
    – quid
    Commented May 14, 2015 at 0:18
  • $\begingroup$ Does anybody have access to Volterra's papers from 1896 "Sulla inversione degli integrale definiti" in R. Acc dei Lincei and R. Acc di Torino? I couldn't find them on-line. What, if anything, does he call the kernel there? $\endgroup$ Commented May 14, 2015 at 6:26
  • $\begingroup$ You might ask this question on History of Science and Mathematics $\endgroup$
    – quid
    Commented May 14, 2015 at 10:04

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