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.

Click here for the paper.

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

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6 Responses to “Convolutions and the Weierstrass Approximation Theorem”

    • Matthew Bond said

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

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

  2. […] 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 […]

  3. […] 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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