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Results tagged with examples
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user 21907
For questions requesting examples of a certain structure or phenomenon
1
vote
Is there a non-trivial example for a 1-homogeneous function satisfying a specific inequality...
Take $A(x)=\vert x\vert$ the Euclidean norm in $\mathbb R^n$, which is obviously homogeneous of degree 1.
We have
$$
A'(x)=\frac{x}{A(x)},\quad A''(x)=\frac{Id}{A(x)}-\frac{x\otimes x}{A(x)^3}=A(x)^{- …
- 16.2k
9
votes
What's the use of Malgrange preparation theorem?
Take $k=1$ in your statement. There are two easy cases: the first one is when $f$ is real-valued, then you have only to use the implicit function theorem to get a normal form $t+a(x)$, up to a unit (a …
- 16.2k
5
votes
Example for the Sobolev embedding theorem when p=n.
Take $u$ in $\mathscr S'(\mathbb R^2)$ with
$$
\hat u(\xi)=\frac{\mathbf 1(\vert\xi\vert\ge 2)}{\vert\xi\vert^3 \ln\vert\xi\vert},\quad
\vert \xi\vert^2\hat u(\xi)=\frac{\mathbf 1(\vert\xi\vert\ge 2)} …
- 16.2k