Universal function approximation with fixed values (as vector or matrix)
By "plain fixed numbers" do you mean a piecewise-constant function?  Sure, those are universal approximators (with some natural hypotheses*).  Indeed, decision trees generate a special kind of piecewise-constant function, and they are universal approximators*.

*I've actually been unable to find references for these, and am not sure how to work out reasonable hypotheses.  Using the examples $\sin(x)$ and $\sin(1/x)$ for $x&gt;0$ indicates that we need a compact domain, and surely some sort of continuity is required.  That's probably enough, but my analysis classes are too long forgotten to know how to produce a proof; you might need some fancier stuff like Stone-Weierstrass.  If you take Lipschitz continuity as a hypothesis, it seems like there'd be an easy(/ier) direct argument.