Talk:Lacunary function
From Wikipedia, the free encyclopedia
[edit] Notes on original article
I started working on this article for two reasons.
- I'm currently reworking the article on complex logarithms, and that article naturally relates to the process of analytic continuation, which in turn relates to lacunary functions – these serve as a sort of brake or limit on the continuation process.
- I already knew a little bit about lacunary functions. When I ran the wiki search to look for "lacunary" I located one red link, on "lacunary series", in the article about Szolem Mandelbrojt. When I read that B. Mandelbrot, a Caltech alum, is his nephew, I grew even more interested.
Anyway, I'm going to add one more bit, about lacunary trigonometric series, and then get back to the complex logarithms, and topics in complex-valued continued fractions. I do have a couple of ideas, though, for any other authors who may come wandering through here.
- The "Hadamard gaps" of size pk have apparently been reduced to the size kp, which is almost the difference between an exponential and a linear function. I'm not sure what the best results are right now for proving the existence of the circle of singularities ... but I did locate some definitions of "(P,A)-lacunary functions" that defined the much smaller "gaps" between non-zero coefficients. From the little bit I was able to read, I remained uncertain whether these more modern "lacunary" functions still stop analytic continuation cold, or if they're a new kind of critter that just took over a conveniently available name.
- There seemed to be a lot of literature on lacunary Fourier series, but I haven't really dug into that any farther than to locate a couple of definitions and one basic theorem for this article. The stuff I did read stressed Lebesgue measure and some stuff like that ... I'm pretty rusty on measure theory.
- Intuitively, the boundary circle of a lacunary function is very chaotic. Here's this smoothly varying function with derivatives of every order in a region, but it blows up all along the boundary. It reminds me, in a way, of the boundary of the Mandelbrot set, and I think it's very interesting that two guys who are closely related both worked on objects that exhibit a kind of chaotic, but predictable, behavior.
Well, that's all I had in mind right now. DavidCBryant 01:35, 19 January 2007 (UTC)
