Approximation in $L^2$ by piecewise constant functions

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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 $h$.

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).

So, does the approximability property of $L^2$ functions by piecewise constant function hold? If yes, what theorem/author can I refer to?

Thanks in advance. Best regards, Sebastien

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3 Answers

A few word about the setting of $L^2$ approximation scheme by simple functions. If $(\Omega,\mathcal{A},\mu)$ is a measure space and $\mathcal{B}$ is a finite sub-algebra of $\mathcal{A}$, generated by the partition $\mathcal{E}$, the class $L^2(\Omega,\mathcal{B},\mu)$ is a finite dimensional subspace of $L^2(\Omega,\mathcal{A},\mu)$ consisting of simple functions which are constant on each element of $\mathcal{E}$. The best approximation of $w\in L^2(\Omega,\mathcal{A},\mu)$ is the simple function that takes the value $\frac{1}{\mu(E)}\int_E vd\mu$ on each set $E\in\mathcal{E}$. Further, any sequence of finite algebras $(\mathcal{B_n})_{n\in \mathbb{N}}$ that generate $\mathcal{A}$ as a $\sigma$ algebra, produce this way an approximation of the identity (the corresponding orthogonal projectors converge strongly to the identity).

In general, you may be interested in the notion of conditional expectation relative to a $\sigma$-algebra (the orthogonal projector).

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It's obvious. Continuous functions are dense in $L^2$, and piecewise constant functions approximate a continuous, even in the uniform metric.

Sorry, for my English.

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Dmitry's answer addresses your question. If you're looking for the rate of convergence in h, then -you need to say a bit more about the regularity of w. I'd recommend the book by Brenner and Scott.

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