MHB Understanding the Integral in a Proof: A Closer Look at the Role of Supp and Chi

  • Thread starter Thread starter evinda
  • Start date Start date
  • Tags Tags
    Integral
evinda
Gold Member
MHB
Messages
3,741
Reaction score
0
Hello! (Wave)

I am looking at the following proof:View attachment 5454

Why does it hold that $\int u_j \phi dx= \int (\psi_j \ast u) \phi dx$ ?
 

Attachments

  • conc.JPG
    conc.JPG
    24.5 KB · Views: 108
Physics news on Phys.org
We have that $X(\frac{x}{j})=1$ for $|x|<j$, right? And since we pick $j$ sufficiently large, it's like if we had $-\infty<j<+\infty$ and so we have $X(\frac{x}{j})=1$ for all the values of $x$, right?
 
You are correct that $\chi(x/j) = 1$ for $|x| < j$. To proceed, let $M > 0$ such that $\operatorname{supp}(\phi)\subset [-M,M]^n$; choose $j > M$. Then $\phi(x) = 0$ for all $|x| \ge j$. Thus, for all $j > M$,

$$\int u_j\phi\, dx = \int \chi\left(\frac{x}{j}\right)(\psi_j * u)\phi(x)\ dx = \int_{|x| < j} \chi\left(\frac{x}{j}\right)(\psi_j * u)\phi(x)\, dx = \int_{|x| < j} (\psi_j * u)\phi\, dx = \int (\psi_j * u)\phi\, dx$$
 
Euge said:
You are correct that $\chi(x/j) = 1$ for $|x| < j$. To proceed, let $M > 0$ such that $\operatorname{supp}(\phi)\subset [-M,M]^n$; choose $j > M$. Then $\phi(x) = 0$ for all $|x| \ge j$. Thus, for all $j > M$,

$$\int u_j\phi\, dx = \int \chi\left(\frac{x}{j}\right)(\psi_j * u)\phi(x)\ dx = \int_{|x| < j} \chi\left(\frac{x}{j}\right)(\psi_j * u)\phi(x)\, dx = \int_{|x| < j} (\psi_j * u)\phi\, dx = \int (\psi_j * u)\phi\, dx$$

I got it... Thanks a lot! (Smirk)
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
Back
Top