I’d like to begin today by welcoming Timothy Gowers to the mathematics blogging community; Tim’s blog will also double as the “official” blog for the Princeton Companion to Mathematics, as indicated by his first post
which also contains links to further material (such as sample articles) on the Companion. Tim is already thinking beyond the blog medium, though, as you can see in his second post

Anyway, this gives me an excuse to continue my own series of PCM articles. Some years back, Tim asked me to write a longer article on harmonic analysis – the quantitative study of oscillation, transforms, and other features of functions and sets on domains. At the time I did not fully understand the theme of the Companion, and wrote a rather detailed and technical survey of the subject, which turned out to be totally unsuitable for the Companion. I then went back and rewrote the article from scratch, leading to this article, which (modulo some further editing) is close to what will actually appear. (These two articles were already available on my web site, but not in a particularly prominent manner.) So, as you can see, the articles in the Companion are not exactly at the same level as the expository survey articles one sees published in journals.

I should also mention that some other authors for the Companion have put their articles on-line. For instance, Alain Connes‘ PCM article “Advice for the beginner“, aimed at graduate students just starting out in research mathematics, was in fact already linked to on one of the pages of this blog. I’ll try to point out links to other PCM articles in future posts in this series.