Talk:Lacunary function

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 p^k have apparently been reduced to the size k^p, 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)

What about a function like f(z)=1+z^3+z^5+z^6+z^7+z^9 ... = 1/(1-z)- (z+z^2+z^4+z^8....). It differs from a lacunary function by a finite amount at points arbitrarily close to all bar one point on the unit circle, so should likewise have a circle of essential singularities there, though the gaps in coefficients are increasingly sparse. —Preceding unsigned comment added by 91.105.133.181 (talk) 22:07, 24 April 2008 (UTC)

I think x^2+x^3+x^5+x^7+x^11+... (with prime numbers as exponents) is also lacunary, although the exponents grow slower than geometrically and Hadamard's theorem cannot help in this case. If you evaluate more and more terms in the case x=i, the result usually heads toward -1 - i*infinity, but sometimes switches back toward -1 + i*infinity, it is an example of Chebyshev's bias which says there seems to be more primes of the form 4k+3 than 4k+1.Blouge (talk) 18:39, 22 May 2008 (UTC)

Yes, the prime power function is lacunary. This follows from the Fabry gap theorem and the prime number theorem. I was about to add the following sentences, but maybe they count as original research? "Even slower gap growth can induce lacunarity, as per the Fabry gap theorem. For example, together with the Prime number theorem this implies lacunarity of $f(z)=\sum _{k=1}^{\infty }z^{p_{k}}$ where $p_{k}$ is the k'th prime." Anders Sandberg (talk) 17:37, 3 January 2017 (UTC)