14
$\begingroup$

I've asked the following question on math.stackexchange but there has been no response so I'll ask it again here:

A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a "diagonal" coefficient of a unitary representation of $G$.

For a definition and discussion of positive definite function see here.

I've often wished I had a collection of diverse examples of positive definite functions on groups, for the purpose of testing various conjectures. I hope the diverse experience of the participants of this forum can help me collect a list of such examples.

To clarify what I'd like to see:

What is an example of a positive definite function on a group $G$ that is not easily seen to be a coefficient of a unitary representation of $G$? What are some positive definite functions that arise in contexts sufficiently removed from studying the coefficients of unitary representations?

Also, the weirder the group $G$ the better. I'd like a collection of quirky beasts...

$\endgroup$
1

3 Answers 3

7
$\begingroup$

For me, often positive-definite functions arise as kernel functions in machine learning

A small list can be found at this link

Also, I would also add one of the classic books on this subject:

Harmonic analysis on semigroups by Christian Berg, Jens Peter Reus Christensen, Paul Ressel.

$\endgroup$
1
  • $\begingroup$ This is exactly the sort of thing I'm looking for, Suvrit. This connection with machine learning is quite new to me! $\endgroup$
    – Jon Bannon
    Aug 22, 2011 at 17:02
6
$\begingroup$

I guess the example below provides one answer to your first question.

A famous positive-definite function** is the one in the Bessis-Moussa-Villani conjecture:

Let $A$ and $B$ be $n \times n$ Hermitian matrices. Then the function $$\phi(t) = \mbox{trace}(e^{A+i t B}),$$ is a positive-definite function.

** Conjectured to be positive-definite, though apparently it has been proved very recently; however, until that has been verified independently, I will adhere to the safety of the word "conjecture"

$\endgroup$
6
$\begingroup$

Perhaps you are already aware of this, but I thought I'd mention it for other interested google-enabled readers.

  1. Infinitely divisible distributions are one place where positive-definite functions come up (Lévy processes, Lévy-Khintchine formula, etc., are also relevant keywords)

  2. Infinite divisibility in Free Probability is another related place.

$\endgroup$
1
  • $\begingroup$ The link to springerlink.com is broken. I'm also unable to find any copy saved on the Wayback Machine. $\endgroup$ Apr 17, 2023 at 5:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.