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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 $d$?

Thanks.

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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

1 Answer 1

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 $C ~ n |I|$.

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