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I'm trying to understand the answer to this MO question: Bochner's theorem for measures of positive type, which suggests a relationship between Bochner's theorem and the Plancherel theorem.

The Bochner-Schwartz theorem states that a distribution is positive-definite if and only if it is the Fourier transform of a nonnegative tempered measure. This seems to hold for locally compact abelian groups.

The Plancherel theorem says—correct me if I am wrong—that there is a measure on the unitary dual of a given locally compact (not necessarily abelian) group, such that the Plancherel and inversion formulas hold, and more.

I'd like to understand better: what does the Plancherel theorem say about positive-definite distributions?

In the paper of Godement cited, he shows how a continuous function of positive type can be written as a transform of the Plancherel measure, so the Plancherel theorem implies a part of Bochner. He also mentions that this works for distributions of positive type. So a positive-definite distribution can be written like so. On the other hand, does the Plancherel theorem give any condition for a distribution to be positive definite, like Bochner-Schwartz does?

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    $\begingroup$ First comment straight out of the box: the Plancherel formula (in its usual interpretation) does not hold for all locally compact groups, you usually need something like the asssumption of the group being Type I. Non-abelian free groups are not Type I; neither is the integer Heisenberg group; but connected semisimple Lie groups are, and probably loads of groups the number theorists care about (e.g. ${\rm GL}(n,{\bf Q}_p)$) are also Type I $\endgroup$
    – Yemon Choi
    Commented Jul 7, 2015 at 16:21
  • $\begingroup$ Thanks for the correction. This seems to be a subtle point I haven't heard mention from number theorists, as you say. $\endgroup$
    – Tian An
    Commented Jul 7, 2015 at 23:22

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