I'm looking for a dense subset of the Nikol'skii space $N^{s,p}(\mathbb{R}^N)=B_{p,\infty}^s(\mathbb{R}^N)$, $s\in(0,1)$, $p\in(1,\infty)$, the definition of which is recalled below:

$N^{s,p}(\mathbb{R}^N):=\left\{f\in L^p:|f|_{N^{s,p}}:=\displaystyle\sup_{h\in\mathbb{R}^N,h\ne0}|h|^{-s}\int_{\mathbb{R}^N}|f(x+h)-f(x)|^p\mathrm{d}x<\infty\right\}$

It is endowed with the (quasi-)norm: $\|f\|_{N^{s,p}}=\|f\|_{L^{p}}+|f|_{N^{s,p}}$.

I've been through the classical books on the topic (Triebel's monographs, etc...) but the best I could find is that neither $C_0^\infty$ nor $\mathscr{S}$ are dense in $N^{s,p}$. What's more, most of the references I could get were in russian... Does anyone know a (possibly simple - if not, I'll take too) class of functions dense in $N^{s,p}$ or any good reference ? Perhaps something like $C^\infty\cap W^{\sigma,p}$ for some $\sigma>0$, or some convolutions of $f\in N^{s,p}$ by appropriate kernels?

  • 2
    $\begingroup$ @Michael Do we really need the tag "modern-analysis"? See meta discussion $\endgroup$ – mrf Dec 3 '15 at 8:47
  • $\begingroup$ It might be a good idea to remind the reader of the definition of $B^s_{p,\infty}$. I suspect that will give you a better chance of getting an answer. $\endgroup$ – mrf Dec 3 '15 at 8:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.