User s&#233;bastien - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:47:16Z http://mathoverflow.net/feeds/user/14712 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63109/approximation-in-l2-by-piecewise-constant-functions Approximation in $L^2$ by piecewise constant functions Sébastien 2011-04-27T03:04:06Z 2011-05-24T00:01:28Z <p>Dear all, in order to prove the validity of my Galerkin approach of a certain variational problem, I need to check the so-called approximability property. In my case, it boils down to showing that for all $w\in L^2(\Omega)$, $\lim_{h\rightarrow 0}\inf_{w^h\in V^h}||w-w^h||=0$, where $\Omega=[0; 1]^d$, and $V^h$ is the space of piecewise constant functions on a regular (orthogonal) grid, with step $h$.</p> <p>It is probably a classical result. However, I've browsed quickly the finite element literature, and the regularity requirements on $w$ are usually stronger ($H^1$ for example).</p> <p>So, does the approximability property of $L^2$ functions by piecewise constant function hold? If yes, what theorem/author can I refer to?</p> <p>Thanks in advance. Best regards, Sebastien</p> http://mathoverflow.net/questions/63087/packing-density-of-randomly-deposited-circles-on-a-plane/63119#63119 Answer by Sébastien for Packing density of randomly deposited circles on a plane Sébastien 2011-04-27T06:20:37Z 2011-04-27T06:20:37Z <p>If you are interested in a physicist's point of view on the question, you might like to look at <a href="http://cherrypit.princeton.edu/book.html" rel="nofollow">Random Heterogeneous Materials, S. Torquato</a>. It is now a little bit dated, but it provides a very extensive list of references.</p>