Let $\Omega$ be an open subset of $\mathbb{R}^n$ and take a family of continuous compactly supported functions $f\_n$ on $\Omega$ normalized to one (in the $L^2$ sense).


Then, these functions span a subspace $V$ of $L^2$. Now, consider two $L^2$ functions $g$ and $h$ with non-intersecting support. Assume that both $g$ and $h$ only overlap with $k$ of those functions $f\_n$. It may happen now that although $\langle g,h \rangle =0$ due to the non-intersecting support of the two, $\langle P\_V g,P\_V h \rangle \neq 0$ where $P\_V$ is the projection onto $V$. However, we expect that "morally" $\langle P\_V g,P\_V h \rangle$ should still be small if they are far apart and the $(f\_n)$ are sufficiently localized.


Of course one could trivially estimate $\langle P\_V g,P\_V h \rangle \le \left\lVert g\right\rVert \left\lVert h \right\rVert,$ but this estimate is very bad in general as it does not capture the fact that most of the time, this should be something close to zero, since $\langle g,h \rangle =0.$


I ask: Does anybody see a better estimate that captures this $\langle P\_V g,P\_V h \rangle \neq 0$ being small? Especially, one would expect that the larger $V$ is, the closer should we get to zero.


Let me rephrase the questions like this: Assuming you are not allowed to evaluate $P\_Vg$ or $P\_Vh$ but only $g,h$ and all $(f\_n)$ and inner-products among those objects. Can we get from this a better bound? 



Compactly supported functions and projections
I think V is large means that V tends to $L^2(\Omega)$, i.e. you can select enough elements $f\_n$ so that $V\_n = span\{f\_1,\cdots,f\_n\}$ and the space span{$f\_1,\cdots,f\_{\infty} $} forms a complete basis of $L^2(\Omega)$. and (it should means that , otherwise you could add infinite k with support has no intersection with g ,h, then $\langle P\_V g, P\_V h \rangle $ is invariant). 


Under this assumption, $\langle P\_V g, P\_V h \rangle $ should tend to 0, since 


$ \langle P\_V g, P\_V h \rangle - \langle g, h \rangle = 
\langle P\_V g, P\_V h \rangle - \langle P\_V g, h \rangle +
\langle P\_V g, h \rangle - \langle g, h \rangle \leq
\| P\_{V^{\bot}} h\| + \| P\_{V^{\bot}} g\|$ 


tends to zero if V is large in the sense V tends to $L^2(\Omega)$.


