1
$\begingroup$

I'm currently reading this paper and the authors define the set $C^{k,1}(\mathbb{R}^n)$ as consisting of all functions $f:\mathbb{R}^n\rightarrow \mathbb{R}$ having $k$ derivatives and for which: $$ \|f\|:= \max_{|\beta|<k}\left[ \sup_{x \in \mathbb{R}^n} |(D^{\beta}f)(x)| + \sup_{x,y;\, x\neq y} \frac{|(D^{\beta}f)(x)-(D^{\beta}f)(y)|}{\|x-y\|} \right]<\infty. $$

However, I'm having trouble understanding non-trivial examples of such functions (e.x.: if $f$ is a polynomial of degree $k-1$.)

So I ask: What are examples of functions in $C^{k,1}(\mathbb{R}^n)$ but not in $C^{k+1}(\mathbb{R}^n)$?*

Specifically, I imagine there is some condition to show that "piecewise polynomial functions of degree at-most $k$" belong to $C^{k,1}$.?

$\endgroup$
4
  • $\begingroup$ Perhaps it would be helpful if you cited the paper you are reading. $\endgroup$
    – efs
    Commented May 27, 2021 at 1:12
  • $\begingroup$ @EFinat-S I added the paper link. $\endgroup$
    – ABIM
    Commented May 27, 2021 at 1:14
  • 1
    $\begingroup$ When $k$ is even: $\frac{1}{1 + |x|^{k+1}}$. $\endgroup$ Commented May 27, 2021 at 1:37
  • $\begingroup$ Shouldn't that read $|\beta| \le k$ below the maximum? $\endgroup$
    – user56029
    Commented May 27, 2021 at 8:14

1 Answer 1

3
$\begingroup$

For $n=1$ and $k=1$, let $$f(x)=x^2 \sin(1/x)$$ for real $x\ne0$, with $f(0)=0$. Then $f\in C^{k,1}(\mathbb R^n)\setminus C^{k+1}(\mathbb R^n)$.

It should be easy to extend this example to the other $n$ and $k$.

$\endgroup$
1
  • $\begingroup$ Thanks, Im wondering if this set contains some sort of "higher-order" splines... $\endgroup$
    – ABIM
    Commented May 27, 2021 at 1:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.