# Meaning of “kernel”

In analysis, there are at least three kinds of "kernel" concepts:

1. In probability theory, there is a concept called transition probability, also called probability kernel, from one measure space $X$ to another $Y$. It is actually a family of measures on $Y$, indexed by members of $X$. It can transform a measure on $X$ to a measure on $Y$, in terms of integration.

A special case is kernel density estimation of probability density function in statistics, where the kernel is actually a probability kernel.

2. In real/complex analysis, there is a concept called kernel function from a product space $X \times Y$ to $\mathbb{R}$ or $\mathbb{C}$. It can transform some special function: $X \to \mathbb{R}$ or $\mathbb{C}$ to another one:$Y \to \mathbb{R}$ or $\mathbb{C}$. This is used for defining integral transformation

3. In Hilbert space theory, there is concept called positive definite kernel, which is a family of bounded mappings for a family of Hilbert spaces. For every two Hilbert spaces $X$ and $Y$ in the family, there are exactly two members of the kernel, one mapping from $X$ to $Y$ and the other from $Y$ to $X$.

I wonder

1. Are they related concepts, since they seem to share some indescribable similarity? Can they be unified?

In particular, the first two are similar to each other as the kernels induce transformations on measures and on functions in terms of integrals.

It is however not obvious to me how the third one, i.e. the Hilbert space one, is related to the second one.

2. Are they related to the algebraic kernel concepts? Or even to the categorical kernel concept?

I feel like some insights and references for catching the general ideas shared across as many kinds of "kernels" as possible.

Thanks and regards!

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Related question on MO: mathoverflow.net/questions/24098/… – Nate Eldredge Mar 29 '12 at 18:52
Words get used to mean different things all the time. See, for example, mathoverflow.net/questions/7389/… . – Qiaochu Yuan Mar 29 '12 at 18:56
You can add to these the perfect kernel that arises in the Cantor-Bendixson theorem. – Dave L. Renfro Mar 29 '12 at 21:26
@DaveL.Renfro: Thanks! Can you give some references for that? I am not able to find something on the internet about it. – Tim Mar 29 '12 at 21:59
Don't forget the squarefree kernel of an integer as seen, e.g., at en.wikipedia.org/wiki/Radical_of_an_integer – Gerry Myerson Mar 29 '12 at 23:57