2018/05/13 by Polina Yu. Glazyrina, Glazyrina, Polina Yu., Szilárd Gy. Révész +1
Mathematics · #52A10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Point processes and geometric inequalities #Primary 41A17. Secondary 30E10 #Spectral Theory in Mathematical Physics #math.CA #msc:30E10 #msc:41A17. #msc:52A10
paper · pdf · doi:10.48550/arxiv.1805.04822
2 figures
arxiv created 2018/05/13 · openalex publication_date 2018/05/13 · arxiv updated 2018/05/15 · openalex created_date 2019/03/11 · openalex updated_date 2026/07/28
In 1939 P. Turán started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm 1 on I:=[-1,1] (interval) and D:=\z∈ℂ~:~|z|≤ 1\ (disk), under the normalization condition that the zeroes of the polynomial in question all lie in I or D, respectively. For the maximum norm he found that with n:=\mathopdeg p tending to infinity, the precise growth order of the minimal possible derivative norm is √(n) for I and n for D. J. Erőd continued the work of Turán considering other domains. Finally, a decade ago the growth of the minimal possible ∞-norm of the derivative was proved to be of order n for all compact convex domains. Although Turán himself gave comments about the above oscillation question in Lq norms, till recently results were known only for D and I. Recently, we have found order n lower estimations for several general classes of compact convex domains, and conjectured that even for arbitrary convex domains the growth order of this quantity should be n. Now we prove that in Lq norm the oscillation order is at least n/log n for all compact convex domains.