4
$\begingroup$

Let $k_1,\ldots,k_n$ be distinct integers. Let $s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt)$ be a trigonometric sum. Consider any interval $I\subset [-\pi,\pi)$ of length $\delta=\delta(n)$. Let $\,U$ be a uniform distribution in the interval $I$. I am interested in the quantity $\mathbb{P}(s_n(U)>0)$.

Questions: 1) (strong form) How large should $\delta(n)$ be so that we would have $\mathbb{P}(s_n(U)>0)\approx 1/2$?

2) (weak form) How large should $\delta$ be so that we would have $\mathbb{P}(s_n(U)>0)$ is bounded away from zero and one independently of $n$?

Comment: If $s_n$ is the Dirichlet kernel, that is, $k_i=i$, it is easy to see that we must have $\delta_n>>n^{-1}$. I would be content if one of the latter statements was true with $\delta(n)=1/\log (n)$.

$\endgroup$
6
  • $\begingroup$ Need to explain what is $f$. $\endgroup$ Oct 21, 2014 at 9:28
  • $\begingroup$ @Liviu Nicolaescu Sorry, $f=s_n$, I corrected the typo. $\endgroup$
    – TOM
    Oct 21, 2014 at 9:42
  • 1
    $\begingroup$ Do you have any particular reason to believe that even, say, $\delta_n=\pi/2$ works for large $n$? $\endgroup$
    – fedja
    Oct 23, 2014 at 0:06
  • $\begingroup$ Take a very fast growing sequence of $k_j>0$. Then $f=2s_n/\sqrt{n}$ has essentially standard normal distribution and $f(x)^2-2+n^{-1/2}f(2x)$ has all coefficients equal. However the probability that the standard normal random variable is between $-\sqrt 2$ and $\sqrt 2$ is not $\frac 12$. This shows that we can skew the sign distribution somewhat at any scale. The question is how much... $\endgroup$
    – fedja
    Oct 23, 2014 at 11:45
  • $\begingroup$ @fedja Could you elaborate the statement about similarity with the normal distributions? $\endgroup$ Oct 23, 2014 at 16:41

1 Answer 1

3
$\begingroup$

enter image description here

Above you see a $12$-frame animation of the family of trig polynomials

$$ P_k(t) =\cos t+\cos 2t+\cos 3t+\cos kt,\;\;\; k=4,\dotsc, 15. $$

I have included it to illustrate the fact that there seems to be another quantity relevant to your question, besides $n$, namely the degree $d=\max\{k_i;\;\;i=1.\dotsc, n\}$. In the above example $n=4$, but the degree varies from $4$ to $15$.

The next animation may be more suggestive because you can see large intervals where the trig polynomials are negative. More precisely, below is a $10$-frame animation of the trig polynomials

$$P_k(t) =\cos 2t+\cos 3t+ \cos(4k+1)t, \;\;k=4,\dotsc, 13. $$

enter image description here

$\endgroup$
4
  • $\begingroup$ 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. $\endgroup$
    – TOM
    Oct 23, 2014 at 2:29
  • $\begingroup$ After looking at many examples, your conjecture seems more and more plausible (and difficult). Fedja.s $\endgroup$ Oct 23, 2014 at 9:06
  • $\begingroup$ 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 $c\in (0,1)$. $\endgroup$ Oct 23, 2014 at 9:13
  • $\begingroup$ the latter is exactly the weaker form I asked (question 2). $\endgroup$
    – TOM
    Oct 23, 2014 at 10:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.