Approximation theorems

Question

The Weierstrass' approximation theorem for continuous functions on a compact space by using polynomials is well-known. As far as I know, there are some variants of this theorem, e.g. Stone-Weierstrass that refers not only to polynomials as approximator functions. Where could I find these Weierstrass-like approximation theorems? On-line references are OK, but one might also point to some books.

Thanks in advance, Lucian

Answer 1

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.

Answer 2

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.

Answer 3

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

Answer 4

For approximation in the complex domain, I recommend Gaier's "lectures on complex approximation", which covers Mergelyan's theorem, Arakelyan's theorem (approximation of functions on closed, but not necessarily compact sets by entire functions) and related results.

Answer 5

If you want some technical challenge (or let's say, it really is for me) you can have a look at A. Pinkus. N-widths in Approximation Theory, Springer-Verlag, New York, 1980.