MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given the polynomial space $\Pi_d^n$ which contains all the d-variable polynomials of degree up to $n$ and some scattered points $X\subset \Omega\subset\mathbb{R}^d$ ($\Omega$ is an open subset).

Since $\Pi_d^n$ is finite dimensional space there must be a constant $C$ such that

$ \left\|p\right\|_{l_2(X)}\leq C\left\|p\right\|_{L_2} $ for all $p\in\Pi_d^n$

where $\left\|p\right\|^2_{l_2(X)} = \frac{1}{|X|}\sum_{x\in X} p(x)^2$ and $\left\|p\right\|^2_{L_2} = \int_{\Omega} p(x)^2dx$.

Is it possible to find the constant $C$? Or at least gauge some understanding into how $C$ depends on $\Omega$, $X$, $n$ and $d$?


share|cite|improve this question
One could also consider $$\|p\|_\mu^2 = \int_\Omega p(x)^2\,d\mu(x)$$ for other Borel probability measures $\mu$ on $\Omega$. You have two extreme cases. Are they extreme? – Gerald Edgar May 1 '11 at 13:05
Not sure my comment is actually helping, but what do you mean by "what C depends on"? I guess $C$ depends on $\Omega$, $X$, $n$ and $d$. – Joël Cohen May 1 '11 at 14:19
It seems more reasonable that $C$ doesn't depend on the exact location of the scattered points $X$ but instead on the fill distance of $X$ in $\Omega$. I was hoping for some idea of how the constant depends on $\Omega$, $X$, $n$ and $d$. – alext87 May 1 '11 at 17:04

I guess it is possible to gain some understanding at least understand rough bounds. I will just consider the case $d = 1$. First observe that $$ \|p\| _{\ell^2(X)} \leq \|p\| _{\infty}. $$ Second we have on can always choose a one point set $X$, depending on $p$ such that equality holds. Next, one can derive for $\Omega$ an interval bounds on $\|p\| _{\infty}$ from $\|p\|_2$ using the inequality $$ |p'(x)| \leq \frac{n}{\sqrt{1 - x^2}} \|p\| _{L^{\infty}([-1,1])} $$ (I hope I remember the constants correctly). Versions for several intervals can be found in an Acta paper by V. Totik (around 2000).

So what one gets in this case is that $C ~ n |I|$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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