Function such that $ \max_{\xi \in [-1,1]} |f^{(n)} (\xi)|$ is not $o\left(2^n (n+1)!\right)$

Question

Learning about interpolation theory, I'm curious about an example of a $C^\infty$ real function $f$ over $[-1,1]$ such that $$ \max_{\xi \in [-1,1]} |f^{(n)} (\xi)|\neq o\left(2^n (n+1)!\right)$$

Actually, I'd be happy if you find such a function defined on any other real interval.

I don't really have any useful thoughts about this...

score 1 · Accepted Answer · edited Apr 13 '17 at 12:20

1

Given any sequence $(a_n)_{n\geq0}$ of real numbers, there is a function $f\in C^{\infty}(\Bbb{R})$, with $f^{(n)}(0)=n! a_n$. See Borel's Theorem. So, there is function $f\in C^{\infty}(\Bbb{R})$ with $f^{(n)}(0)=n^n n!$ for example. This answers your question.

edited Apr 13 '17 at 12:20

Community

1

answered May 12 '14 at 22:54

Omran Kouba

28,772

That's nice to know about, thanks. Do you know where I can find a proof of this theorem (French and English literature are alright).? – Gabriel Romon May 12 '14 at 23:01
There are references in the above link, or may be you can look at http://www.math.ubc.ca/~cass/research/pdf/Emile.pdf – Omran Kouba May 13 '14 at 06:08
I actually found an easy one in Routières Petit guide de calcul différentiel. – Gabriel Romon May 13 '14 at 08:57

Function such that $ \max_{\xi \in [-1,1]} |f^{(n)} (\xi)|$ is not $o\left(2^n (n+1)!\right)$

1 Answers1