Bounding the discrete l2 norm for polynomials. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:10:56Z http://mathoverflow.net/feeds/question/63593 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63593/bounding-the-discrete-l2-norm-for-polynomials Bounding the discrete l2 norm for polynomials. henryreed 2011-05-01T09:00:07Z 2011-05-02T00:29:56Z <p>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). </p> <p>Since $\Pi_d^n$ is finite dimensional space there must be a constant $C$ such that </p> <p>$\left\|p\right\|_{l_2(X)}\leq C\left\|p\right\|_{L_2}$ for all $p\in\Pi_d^n$</p> <p>where $\left\|p\right\|^2_{l_2(X)} = \frac{1}{|X|}\sum_{x\in X} p(x)^2$ and $\left\|p\right\|^2_{L_2} = \int_{\Omega} p(x)^2dx$.</p> <p>Is it possible to find the constant $C$? Or at least gauge some understanding into how $C$ depends on $\Omega$, $X$, $n$ and $d$?</p> <p>Thanks.</p> http://mathoverflow.net/questions/63593/bounding-the-discrete-l2-norm-for-polynomials/63661#63661 Answer by Helge for Bounding the discrete l2 norm for polynomials. Helge 2011-05-02T00:29:56Z 2011-05-02T00:29:56Z <p>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).</p> <p>So what one gets in this case is that $C ~ n |I|$.</p>