Bounds For Polynomials With A Unit Discrete Norm

Author: E. A. Rakhmanov

Field: Mathematics

Abstract

This paper investigates bounds for polynomials of degree up to n, denoted as P_n, within a set of N equidistant points in the interval (−1, 1). The study focuses on polynomials where the maximum absolute value on this set of points is less than 1. A key result establishes an upper bound for the function K_n,N(x), which represents the maximum of |P(x)| for polynomials in P_n with a unit discrete norm, under the condition that n < N. The obtained bounds are shown to be essentially sharp and are uniform for n < N. The paper presents a general method for deriving these bounds, applicable to more generalized assumptions on the set of points. The analysis delves into the structure of the method, particularly examining the case of equally spaced points. Furthermore, the research explores estimates for K_n,N(x) on the interval [−1, 1] and its generalization to the complex plane. The methodology involves constructing extremal polynomials and utilizing potential-theoretic concepts. The paper also discusses related open problems and conjectures in the field of polynomial approximation.

Table of Contents

• 1. Introduction
• 1.1. Main results of the paper
• 1.2. Outline of the method
• 1.3. Some related open problems for interval and circle
• 2. Proof of Theorem 1
• 2.1. Auxiliary results
• 2.2. Proof of Theorem 1
• 2.3. Proof of the estimate (1.8) and its generalization for x ∈ C
• 3. Proofs of Lemma 1 and Lemma 2
• 3.1. Phase-Amplitude representation of a polynomial with real zeros