Approximation theorems

For approximation with Polynomials, a Weierstrass like theorem is the Muntz's Theorem.

Moving away from polynomials, we have the classic Fourier Series. The Generalization of Fourier series gives rise to many approximation schemes.

Sorry, I wasn't able to find a single page...

Hope that helps.

Functions belonging to reproducing kernel Hilbert spaces can be approximated by weighted discrete sums of the reproducing kernels evaluated at discrete points of the dual variable.

See the following two articles: article-1 article-2.

Perhaps one should mention Runge's theorem, and Mergelyan's theorem which deal with approximation by rational functions and polynomials, respectively.