Functions with compact support

Tuesday, March 14. 2006

Functions with compact support

I finished chapter IV on friday and continued on Hilbert space theory. Unfortunately, when it came to connections with Fourier transformation (Sobolev and Schwartz spaces, distributions), my understanding began to slow down. Some questions occured:

Why is $\langle f^\prime,\varphi\rangle=-\langle f,\varphi^\prime\rangle$ for $\Omega\subset\mathbb{R}^n, \, f\in\mathrm{C}^1(\Omega)$ and $\varphi\in\mathscr{D}(\Omega)$ , i.e., why is

$\displaystyle\int_\Omega\, f^\prime(x)\overline{\varphi(x)}\,dx=-\int_\Omega\, f(x)\overline{\varphi^\prime(x)}\,dx$

for $\varphi$ with compact support? Why do the border terms in the partial integration disappear in this case?
What’s so special about functions with compact support anyway?

Any literature I seeked just had written that fact down without justifying it, as if it were so simply to see. And finally, I saw it: The confusion I had was that I misinterpreted $\varphi$ as being $\mathrm{C}_0(\Omega)$ only, i.e., disappearing at infinity (The closure of $\{x\in\Omega\quad:\quad\varphi(x)\geq\varepsilon\}$ is compact). Here, it definitely was not clear that its values already disappear before infinity.

The rule of partial integration says, $\textstyle\int_a^b f^\prime g = fg|_a^b - \int_a^b fg^\prime$ , where $fg|_a^b = f(b)g(b)-f(a)g(a)$ . If the integration should go over the whole space, then $\textstyle\int_{-\infty}^\infty = \lim_{a\to\infty}\int_{-a}^a$ , and the same is with $fg|_{-\infty}^\infty$ .

Now, if $\varphi\in\mathscr{K}(\Omega)$ with $\Omega\subset\mathbb{R}$ being an interval, it follows that $\exists a,b\in\mathbb{R}$ where $\varphi$ already disappers, because it has compact (and therefore bounded) support; $(a,b)\supset[c,d]$ , where $c$ and $d$ are the left and right borders of $\operatorname{supp}\,\varphi$ . And this is why $f\varphi|_\Omega=0$ in the first question above.

For the sake of completeness, it is clear by the following what $f|_\Omega$ should mean for $\Omega\subset\mathbb{R}^n$ :

$\displaystyle\int_{[a,b]\times[c,d]}f(x,y) \,d(x,y) =$

$\begin{eqnarray*}\displaystyle\quad &=& \int_c^d\int_a^b f(x,y)\,dx\,dy\\ &=& F(x,y)|_{x=a}^{x=b}|_{y=c}^{y=d}\\ &=& \left(F(b,y)-F(a,y)\right)_{y=c}^{y=d}\\ &=& F(b,d)-F(b,c)-F(a,d)+F(a,c)\end{eqnarray*}$

Posted by Stephan Paukner in Master's Thesis at 08:54 | Comments (0) | Trackbacks (0)

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