diff --git a/contents/coarse.tex b/contents/coarse.tex index 98f0ac6692fc552e09a5cdd4d049912952c53e3c..7f3314200c079a6fe9c88d96fc3cde351c5b80a9 100644 --- a/contents/coarse.tex +++ b/contents/coarse.tex @@ -90,7 +90,7 @@ The next main player of coarse index theory is the so-called \emph{localisation In the same setting as above we now look at the family $\varphi \enbrace*{\frac{1}{t} D}$ for $t \in [1,\infty)$.\todo{in what sense is this \enquote{local}?} If $\supp \widehat{\varphi} \subseteq (-r,r)$ we once again get $- \varphi \enbrace*{\frac{1}{t} D} = \frac{1}{\sqrt{2 \pi_n}} \int\limits_{-r}^r \widehat{\varphi}(\xi) e^{i \xi \frac{1}{t} D} \, \mathd \xi = \frac{1}{\sqrt{2 \pi_n}} \int\limits_{-r/t}^{r/t} \widehat{\varphi}(t \cdot \eta) e^{i \xi \eta D} \, \mathd \xi + \varphi \enbrace*{\frac{1}{t} D} = \frac{1}{\sqrt{2 \pi}} \int\limits_{-r}^r \widehat{\varphi}(\xi) e^{i \xi \frac{1}{t} D} \, \mathd \xi = \frac{1}{\sqrt{2 \pi}} \int\limits_{-r/t}^{r/t} \widehat{\varphi}(t \cdot \eta) e^{i \xi \eta D} \, \mathd \xi$ \begin{definition}