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I am reading his proof for Wiener's theorem in Chp9 of Rudin's functional analysis. The theorems (9.4, 9.5 and 9.7) themselves are quite clear and Rudin did a good job explaining the intuition behind them.

The proof for the two lemmas (9.2 and 9.3), however, are rather mysterious to me. I wonder how wiener thought of these two lemmas and saw their connection with the theorems. Are they related to some other theories?


The two lemmas are: (9.2) Let $f\in\mathcal{L}^1$, $t\in\mathbb{R}^d$ and $\epsilon>0$ be given. There is $h\in\mathcal{L}^1$ and $\delta>0$, such that $\|h\|<\epsilon$ and $\hat{h}(s)=\hat{f}(t)-\hat{f}(s)$ for all $s\in B_{\delta}(t)$.


(9.3)Let $Y$ be a closed subspace of $\mathcal{L}^1$. Define $Z(Y)$ to be $\{s\in\mathbb{R}^d:\hat{f}(s)=0 \forall f\in Y\}$. If $\phi\in \mathcal{L}^{\infty}$ satisfies $f\ast\phi=0$ $\forall f\in Y$, then $support(\hat{\phi})\subset Z(Y)$.

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Could you type out the two lemmas? – Alex R. Oct 8 '12 at 17:37
@Alex sure. Please see the edit. – Hui Yu Oct 9 '12 at 2:23

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