# Approximation theorems

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.

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Have you tried Wikipedia? It has a nice article on the Stone-Weierstrass theorem including some references, to Rudin's books among others, and they're really good books. – Alon Amit Aug 25 '10 at 8:58
The book "Studies in Modern Analysis", Math. Assoc. America 1962, edited by McShane, has a paper by Stone himself called "A Generalized Weierstrass Approximation Theorem". According to a review of this book, this paper is a reprint (probably of Stone's 1948 paper in Math. Magazine, but I haven't checked that). Anyway, Stone explains in a leisurely way extensions of Weirestrass's classical theorem to several different settings and gives varied applications. Of course this reference is dated, but you can't go wrong by seeing what Stone had to say. – KCd Aug 25 '10 at 16:09

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.

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Since the MathWorld explanation of "generalized Fourier series" is a bit longish, I'll summarize: in those sorts of series, instead of using sines and cosines, one uses a "basis set" whose "basis functions" satisfy an orthogonality condition. Thus you can have a series in Legendre polynomials (Fourier-Legendre), a series in Bessel functions (though these are more often referred to as Fourier-Hankel series)... and so on. – J. M. Aug 25 '10 at 15:44
I didn't know about Muntz' theorem, thank you. – lmsasu Aug 26 '10 at 7:35
I like Muntz's theorem. – timur Dec 25 '10 at 4:13

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.

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

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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.

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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.

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