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An Extremal Problem for Polynomials

An Extremal Problem for Polynomials R. R. HALL 1. Introduction In this note I prove the following result. THEOREM. Let p(z) be a monk polynomial degree N all of whose roots lie on the unit circle, and E be any sub-set of[—n, n] of measure 2a, 0 < a ^ n. Then I-CO.2). (i) moreover the lower bound given on the right is the best possible. In §2 below I show that for a more general problem, there is an extremal E which, regarded as a sub-set of the torus T = U/2nZ, is connected, i.e. is an interval of length 2a which we may assume (after rotation of the roots of p) to be [ — a, a]. In §3 the minimization over p is reduced to the extremal problem inf P P Jo where P is now a real monic polynomial of degree [N/2] and dji is supported on [cos£a, 1]. It varies according as N is odd or even. The extremal P is determined explicitly in each case. 2. The extremal subsets LEMMA 1. Let y/:[0,2^] -* U be continuous and non-decreasing, and i0 y,(\p(e )\dO (2) where the infimum is taken over the polynomials p and sets E specified in http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

An Extremal Problem for Polynomials

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/17.5.463
Publisher site
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Abstract

R. R. HALL 1. Introduction In this note I prove the following result. THEOREM. Let p(z) be a monk polynomial degree N all of whose roots lie on the unit circle, and E be any sub-set of[—n, n] of measure 2a, 0 < a ^ n. Then I-CO.2). (i) moreover the lower bound given on the right is the best possible. In §2 below I show that for a more general problem, there is an extremal E which, regarded as a sub-set of the torus T = U/2nZ, is connected, i.e. is an interval of length 2a which we may assume (after rotation of the roots of p) to be [ — a, a]. In §3 the minimization over p is reduced to the extremal problem inf P P Jo where P is now a real monic polynomial of degree [N/2] and dji is supported on [cos£a, 1]. It varies according as N is odd or even. The extremal P is determined explicitly in each case. 2. The extremal subsets LEMMA 1. Let y/:[0,2^] -* U be continuous and non-decreasing, and i0 y,(\p(e )\dO (2) where the infimum is taken over the polynomials p and sets E specified in

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Sep 1, 1985

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