3
votes
1answer
112 views

What is the use of $H_s$ for non-integer $s$?

So we have the whole set of theory for Sobolev spaces \begin{equation} H_s(\mathbb{R}^d)=\{u\in D'(\mathbb{R}^d):(1+|y|^2)^{s/2}\hat{u}\in\mathcal{L}^2(\mathbb{R}^d)\}, \end{equation} and we know that ...
2
votes
1answer
56 views

So $k^2-\Delta: H_{s+2}\to H_{s}$ is a homeomorphism, but what does that tell us?

For each $t\in\mathbb{R}$, we define the Sobolev space \begin{equation} H_t=\{u\in\mathcal{S}':\int(1+|y|^2)^t|\hat{u}(y)|^2dy<+\infty\}, \end{equation} where $\mathcal{S}'$ is the space of ...
0
votes
1answer
92 views

How does a myopic interpret Wiener's Tauberian?

I just read about this post on the intuition behind convolution. In Terence Tao's answer convolution is interpreted as the blur of image in near-sighted eyes. In Harald Hanche-Olsen's it is made ...