## Convolutions and the Weierstrass Approximation Theorem

### May 9, 2009

I have a new short paper which proves the Weierstrass approximation theorem in a manner quite similar to how it is done in Rudin’s *Principles of Mathematical Analysis*, but with much more motivation for the technique. In particular, the underlying idea, the convolution (by an approximate identity) is given center stage rather than relegated to the status of a magic trick. **If you have read and vaguely understood the details of Rudin’s proof but are mystified as to how anyone could have thought of that or what the main idea is, then I hope that this paper can help you.**

This paper is probably a bit advanced for a level 2 paper, but in principle, the prerequisites aren’t too intense, and I have some hope that an undergrad could get something from this without having taken Advanced Calc first – maybe a little help from an advanced friend or a professor might be necessary in this case.

Edit: Version 2-0, with some introductory harmonic analysis and Weierstrass for trig polynomials. (8/15/09)

There is this tricki article on the same subject

http://www.tricki.org/article/To_make_a_function_nicer_without_changing_it_much_convolve_it_with_an_approximate_delta_function

That’s good to know, thanks. I tried to be slightly more explicit in some places, but the content is about the same.

[…] By Matthew Bond Convolutions and the Weierstrass approximation theorem has been updated and expanded. Now it also covers the “other” Weierstrass approximation […]

[…] and again By Matthew Bond I seem to be having trouble keeping the mistakes from creeping into my short introduction to convolutions. I had to make a couple corrections, and then I had to make a couple more again today. Most of them […]

[…] Convolutions and the Weierstrass Approximation Theorem – intended for an undergraduate math major audience; experience with epsilon-delta arguments (elementary real analysis) is almost necessary. […]

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