MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Bochner's theorem (for the real line version) asserts an infinite tower of inequalities, as a positivity condition. Taking each one, what do they mean, in an elementary fashion (at least at the start)?

For instance, the $1 \times 1$ matrix says that $Q(0)$ is positive. The $2 \times 2$ says that $|Q(x)| \ge |Q(0)|$. (And these two are commonly written down for necessary conditions of characteristic functions.) What about 3 and 4?

share|cite|improve this question
Bochner (God rest his soul) proved more than one theorem. Perhaps you could state the result you are alluding to? – Igor Rivin Apr 25 '12 at 20:08
I've added a link... – genneth Apr 26 '12 at 9:32
What is $f$? The Wikipedia article you linked to starts with a finite positive Borel measure $\mu$ on $\mathbb R$, takes the Fourier transform to form $Q$, complex-valued a function on $\mathbb R$, and then forms a kernel $K$-- the theorem is that this is positive definite, so for any $n$, given $x_1,\cdots,x_n\in\mathbb R$, the matrix $(K(x_i,x_j)) = (Q(x_j-x_i))$ is positive (semi-)definite. You're asking, I think-- how to I interpret this for a fixed n? So I think your $f$ is Wikipedia's $Q$? – Matthew Daws Apr 26 '12 at 13:40
@MatthewDaws: that is correct. I will change the notation I have used to match the Wiki. – genneth Apr 26 '12 at 16:16

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.