The complex analysis involved is relatively elementary, but intricate. But at any rate, Dr. Alexander Volberg and I now have an analog of the Nazarov, Peres, Volberg result, adapted to the Sierpinski gasket with a moderate loss.

Here it is. I have it labelled as a Level 3, but this one will probably keep you up late regardless of who you are or how much you know.


Comprehensive exam

May 9, 2009

My comprehensive exam was last week. It was arranged so that the timing would coincide with Dr. Isabella Laba‘s visit to MSU, since her research has been related to my own lately.

Before her visit, I had to take my dog Malty to the vet. As usual, he shed a lot on the way there, on the way back, and during the visit. The next day, I got a surprise call from my advisor saying that Dr. Laba’s plane had been delayed, and that I had to pick her up from the airport.

I passed my comprehensive exam (on this paper), but in the evaluation, a remark was anonymously made that I need to vacuum my car. I think I can plead circumstances, maybe.

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)