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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?

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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
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I've added a link... –  genneth Apr 26 '12 at 9:32
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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
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