The version of Wiener's lemma that I know, from Katznelson's Introduction to Harmonic Analysis, is $\lim_{N\to\infty}\frac{1}{2N}\sum_{-N}^{N} |\hat{\mu}(n)|^2 =\sum_{t\in\mathbb{T}}|\mu(\{t\})|^2$. This is about the group $\mathbb{Z}$ and its dual $\mathbb{T}$. Can this be extended to other groups? I expect/hope this is true for locally compact abelian groups but I have no luck when I search. Wiener has many different lemmas and theorems and formulas, so searching is difficult for me. Any references would be greatly appreciated!

By the way, there is this posting on MO where they claim it is true for $\mathbb{R}$: https://mathoverflow.net/questions/64173/a-complex-borel-measure-whose-fourier-transform-goes-to-zero

  • $\begingroup$ I'm sure Katznelson does it for $\mathbb R$. It depends on $\mu *\bar {\mu}(0) = \sum \mu(t)^2$ and $\hat {\mu * \bar{\mu}} = \vert \hat {\mu} \vert^2$ where $\bar {\mu}(x) = \mu(-x)$ ... sec 6.2.9 of Katznelson $\endgroup$ – mike Jun 7 '12 at 11:41

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