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Minima of Cosine Sums and Maxima of Polynomials on the Unit Circle
Author(s) -
Odlyzko A. M.
Publication year - 1982
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-26.3.412
Subject(s) - maxima and minima , integer (computer science) , combinatorics , maxima , mathematics , unit (ring theory) , value (mathematics) , discrete mathematics , mathematical analysis , computer science , statistics , art , mathematics education , performance art , art history , programming language
This note considers the problem of minimizing, under various restrictions on the nonnegative numbers b k , the absolute value of min∑ k = 1 ∞b k cos k k θ. For example, if it is required that exactly n of the b k be ⩾ 1 and all others 0, then this minimum can be as small as O ( n 1/3 (log n ) 1/3 ). Other variants of this problem are used to improve estimates on a problem of Erdös and Szekeres. It is shown that for any positive integer n , there are positive integers a l , …, a n such that max|∏ k = 1 n ( 1 − z a k )|| z | = 1= exp { O ( n 1 / 3( log n )4 / 3)} .