I am somewhat intrigued by the ideas behind approximation theory. So, I would like to know (1) what are some thorough clear reference books to get acquainted with approximation theory; (2) what are the necessary prerequisites.
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- $\begingroup$ In particular, I would like some suggestions specific to Korovkin-type approximation theory. $\endgroup$Dal– Dal2014-11-01 23:26:18 +00:00Commented Nov 1, 2014 at 23:26
- 1$\begingroup$ Try Analysis II in the Encyclopaedia of Mathematical Sciences. $\endgroup$Marcus Johnson– Marcus Johnson2014-11-20 17:51:55 +00:00Commented Nov 20, 2014 at 17:51
- $\begingroup$ @MariusKempe Thank you very much. $\endgroup$Dal– Dal2014-11-20 21:51:47 +00:00Commented Nov 20, 2014 at 21:51
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1 Answer
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Much of approximation theory involves polynomials and approximation underneath the supremum norm. A good understanding of the Weierstrass theorem from the 1800's is a good place to start. In particular, look up Berstein's constructive proof of the Weierstrass theorem. In this vein there is also the Stone-Weierstrass theorem. This is all on the level of an undergraduate Real Analysis class. You can find some details in Rudin, and G. G. Lorentz's book "Bernstein Polynomials."
The next step would be to investigate Fourier series as function approximators. These work well for approximation under the L^2 norm, and when the function is what is called band limited, you can reproduce the function exactly using the shannon sampling theorem. Any book on Fourier series would cover this.
After this you can look into approximations in terms of Wavelets, the book by Walter and Shen "Wavelets and Other Orthogonal Systems" is a really good reference for mathematicians.
Another direction is RKHSs. Here the hilbert space norm dominates the supremum norm, which makes them ideal candidates for function approximation. The book "Support Vector Machines" by Steinwart and Christmaan (Sp?) is a good introduction. (In particular Chapter 4). My university subscribes to SpringerLink, and I can download this from the Springer website for free. You should see if your university has this as well.
The website Survey's in Approximation Theory is a great resource. Here you can find summaries of different approximation methods. Including using Chebychev polynomials and other orthogonal systems. http://www.math.technion.ac.il/sat/index.html
Finally, on arxiv you can find a survey paper on the Korovkin approximation theory: http://arxiv.org/pdf/1009.2601.pdf This has a self contained introduction, and you can find further references inside.
