# Universal function approximation with fixed values (as vector or matrix)

I was thinking about way to represent/approximate universal function and came up with the idea that a plain fixed numbers could be used to represent pretty much any function on a fixed interval.

I wonder how such approach called. And if it's used in logistic regression, genetic algorithms etc.?

The good thing is that it's universal, you can extend it to multiple dimensions and you don't have to make any assumption about function if it should be polynomial or trigonometric etc.

I would like to know more about it. But can't find much. *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 $$x>0$$ 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.