# Bochner's theorem, in stages

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 $$|Q(x)| \le |Q(0)|$$ (These two are commonly written down for necessary conditions of characteristic functions). What about 3 and 4?

• 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

I believe the $$3\times3$$ inequalities were first exploited by Kreĭn and Weil, who wrote \begin{align} \bigl|\,Q(g) - Q(h)\,\bigr| &\leqslant \sqrt{2\mathrm{Re}\bigl(1 - Q(g^{-1} h)\bigr)}, \tag K\\[1ex] \bigl|\,Q(gh) - Q(g)Q(h)\,\bigr| &\leqslant \sqrt{1 - |Q(g)|{}^2}\sqrt{1 - |Q(h)|{}^2}. \tag W \end{align} in (1940), resp. (1940, p. 57). I stated them for any positive-definite function $$Q$$ on any group (written multiplicatively, as none of this requires Bochner’s commutative setting), but normalized $$Q(e)=1$$; else rescale by $$Q(e)$$, which (as you noted) is nonnegative by the $$1\times1$$ case.
Kreĭn’s $$(\mathrm K)$$ says that $$Q$$ is uniformly continuous everywhere as soon as it’s continuous at $$e$$, and I like to think of Weil’s $$(\mathrm W)$$ as a group version of Heisenberg’s uncertainty inequalities. Deducing them from positive-definiteness is not so obvious: $$(\mathrm W)$$ is a rearrangement of the determinant criterion $$\begin{vmatrix} 1&Q(a^{-1}b)&Q(a^{-1}c)\\ \overline{Q(a^{-1}b)}&1&Q(b^{-1}c)\\ \overline{Q(a^{-1}c)}&\overline{Q(b^{-1}c)}&1 \end{vmatrix} \geqslant0, \tag{*}$$ written with $$g=a^{-1}b$$ and $$h=b^{-1}c$$; and proving $$(\mathrm K)$$ from $$(\mathrm W)$$ alone is possible but not very enlightening. I prefer to observe that positive-definiteness of $$Q$$ means positive-definiteness of the sesquilinear form $$(c,d)\mapsto Q(c^*\cdot d)$$ on the group *-algebra $$\mathbf C[G]=$$ {functions $$c:G\to\mathbf C$$ with finite support} where product, *-operation, and linear form $$Q$$ are defined on the basis of (Kronecker) $$\delta^g$$’s by $$\delta^g\cdot\delta^h=\delta^{gh}, \qquad (\delta^g)^*=\delta^{g^{-1}}, \qquad Q(\delta^g) = Q(g).$$ Then $$(\mathrm K)$$ and $$(\mathrm W)$$ are just the Cauchy-Schwarz inequality $$|Q(c^*\cdot d)|^2\leqslant Q(c^*\cdot c)Q(d^*\cdot d),$$ written out for $$(c^*,d) = (\delta^e,\delta^g-\delta^h)$$, resp. $$(\delta^g-Q(g)\delta^e,\delta^h-Q(h)\delta^e)$$.
(Which triples give $$(\mathrm K)$$ and $$(\mathrm W)$$ can then be found by remembering that Cauchy-Schwarz amounts to writing $$Q(f^*\cdot f)\geqslant 0$$ where $$f=Q(c^*\cdot c)d-Q(c^*\cdot d)c$$. Hewitt-Ross (1970, pp. 255, 289) have another proof, and the attribution to Kreĭn & Weil. It might be interesting to know if the $$4\times4$$ version of $$(*)$$ has some nice consequences...?)