Timeline for Chances for a cosine polynomial to be positive at a point

8 events

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Oct 23, 2014 at 10:46	comment	added	TOM		the latter is exactly the weaker form I asked (question 2).
Oct 23, 2014 at 9:13	comment	added	Liviu Nicolaescu		Fedja's question is an excellent place to start. It suggests that maybe the $1/2$ in your question should be replaced by some universal number $c\in (0,1)$.
Oct 23, 2014 at 9:06	comment	added	Liviu Nicolaescu		After looking at many examples, your conjecture seems more and more plausible (and difficult). Fedja.s
Oct 23, 2014 at 2:29	comment	added	TOM		I can't prove it, but I expect "highly structured" examples to be the worst- the ones with lengthy arithmetic progressions ${k_1,\ldots k_n}$. And then degree plays no part. It may be that I am wrong, but as I said, I expect good behavior in sufficiently long intervals. Say $\delta=1/\log(n)$. For $n$ large, say $n=100$ and $\delta=1/\log(100)\approx 0.2$ I would expect many roots in the interval and thus many intervals many sing changes. For $n=4$ one cannot see this effect as the length $\delta$ is comparable to the interval length where the sum of cosines changes sign.
Oct 22, 2014 at 17:05	history	edited	Liviu Nicolaescu	CC BY-SA 3.0	added 43 characters in body
Oct 22, 2014 at 16:02	history	edited	Liviu Nicolaescu	CC BY-SA 3.0	added 288 characters in body
Oct 21, 2014 at 18:12	history	edited	Liviu Nicolaescu	CC BY-SA 3.0	added 5 characters in body
Oct 21, 2014 at 13:43	history	answered	Liviu Nicolaescu	CC BY-SA 3.0