## Update

### August 15, 2009

Convolutions and the Weierstrass approximation theorem has been updated and expanded. Now it also covers the “other” Weierstrass approximation theorem, taking a one-page tour of selected topics from the first chapter of a harmonic analysis text along the way for motivation. Also, a solution to the heat equation in one dimension has been briefly provided without proof in the remarks section.

So now you all should know what a convolution is and have some idea that they’re important.

In other news, a summary of the noodle paper is under review at Comptes Rendus Mathematique. If they choose to publish it, I’d like to link it here and say a few words about it having to do with some recent thoughts of mine concerning the lacunary circular operator, as well as possible connections to Furstenburg sets and incidence geometry. In particular, a comparison of the solution to the joints problem with the estimate taken from the multilinear Kakeya conjecture indicates to me that there is some reason to think that maybe incidence geometry won’t provide the answer to the sharpness of the LloglogL bound on the lacunary circular operator. I’d like to say something at length, but I want to share the paper with you first.