# Number of simple functions needed to approximate another function

I have got a function $$f$$ which maps from one compact space to another. Function $$f$$ is smooth.

I want to approximate it with some simple functions (e.g polynomials). Is there any theory that gives an upper bound for the whole class of smooth functions on what order polynomials one needs to use to make approximation $$\epsilon$$ close to the original function?

I will be grateful for any related literature, maybe not only smooth functions, but anything related to my question

• Are you familiar with the Stone-Weierstrass theorem? – Theo Bendit Jun 4 at 5:34