Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\mathbb{R}$: http://mathoverflow.net/questions/64173/a-complex-borel-measure-whose-fourier-transform-goes-to-zero

share|improve this question
    
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 –  mike Jun 7 '12 at 11:41

Your Answer

 
discard

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

Browse other questions tagged or ask your own question.