Timeline for Polynomials having all their zeros on the unit circle
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 1, 2023 at 17:48 | comment | added | fedja | @Conrad Thanks. I'll still spend some time trying to figure it out myself before looking up the proof in the book and comparing it with the ideas I currently have but it is nice to know that the conjecture is known to be correct :-). BTW, once you know the literature so well, is it also well known what is the best constant in the Bernstein inequality $\|P'\|_{L^p}\le Cn\|P\|_{L^p}$ for non-negative real-valued trigonometric polynomials of degree $n$? I once figured out the case $p=1$ (the extremizer is $1+\cos nx$, of course), but I still don't know the truth for any other $p\ne2,\infty$. | |
| Nov 1, 2023 at 17:28 | comment | added | Conrad | @fedja the result is true but nontrivial with the equality cases determined (they are essentially coming from complex polynomials of the type $c(z^m+e^{ia})^2, a\in \mathbb R$) - not sure who proved this first but a (4 page) proof is presented in the comprehensive book Analytic Theory of Polynomials by Rahman and Schmeisser, Th $16.1.9$ pages 646-650; see global.oup.com/academic/product/… | |
| Nov 1, 2023 at 17:07 | comment | added | GH from MO | @fedja I have not seen this generalization, but I am not familiar with this topic either. I usually cover Visser's inequality in the first week of my complex analysis class, but I did not know until this MO post that the result was due to Visser from 1945. I only knew the problem from the 1991 Schweitzer Contest. | |
| Nov 1, 2023 at 13:17 | comment | added | fedja | @GHfromMO The problem boils down (actually, is equivalent) to the following innocently looking question: if $P=a_0+\Re\sum_{k=1}^n a_kz^k$ is a real trigonometric polynomial of degree $n$ with $2n$ zeroes on the unit circle. Then $|a_0|\le \frac 12\|P\|_\infty$ (or, in other words, the only case of interest is the inequality $\|P\|_\infty\ge 2|a_m|$ for algebraic polynomials $P$ of even degree $n=2m$ with complex coefficients and all zeroes on the unit circle). Have you seen this generalization of Visser anywhere? | |
| Oct 29, 2023 at 16:55 | comment | added | GH from MO | Visser's inequality has been generalized by several researchers (e.g. there was an immediate generalization by Visser and van der Corput), and Google finds many references. Visser's original publication is C. Visser: A simple proof of certain inequalities concerning polynomials, Nederl. Akad. Wetensch. Proc. 48, 276-281 = Indag. Math. 7 (1945), 81-86. | |
| Oct 29, 2023 at 13:51 | comment | added | Conrad | @user by using different roots of unity on $P$ but also on $zP$ and such as well as on the $z^n\bar P(1/\bar z)$ transform clearly one can obtain various similar inequalities as long as we can get rid of all but at most two terms so I am convinced this should be known, but it is easy to prove anyway; similarly the coefficient relation for polynomials with unit circle roots is well known | |
| Oct 29, 2023 at 10:30 | vote | accept | user159888 | ||
| Oct 29, 2023 at 10:27 | comment | added | user159888 | This is well known Visser’s inequality. | |
| Oct 29, 2023 at 7:29 | comment | added | Emil Jeřábek | So $|a_0|+|a_n|$ is a lower bound for all polynomials without any assumptions. | |
| Oct 29, 2023 at 0:54 | comment | added | GH from MO | The statement (in slightly sharper form) was the 2nd problem in the 1991 Schweitzer Contest in Hungary. See Problem G.47 in Székely (ed.): Contests in Higher Mathematics - Miklós Schweitzer Competitions (1962-1991). | |
| Oct 29, 2023 at 0:12 | history | edited | Conrad | CC BY-SA 4.0 |
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| Oct 28, 2023 at 23:48 | history | answered | Conrad | CC BY-SA 4.0 |