An explicit class of min–max polynomials on the ball and on the sphere

Abstract

Let Π n + m − 1 d denote the set of polynomials in d variables of total degree less than or equal to n + m − 1 with real coefficients and let P ( x ) , x = ( x 1 , … , x d ) , be a given homogeneous polynomial of degree n + m in d variables with real coefficients. We look for a polynomial p ∗ ∈ Π n + m − 1 d such that P − p ∗ has least max norm on the unit ball and the unit sphere in dimension d , d ≥ 2 , and call P − p ∗ a min–max polynomial. For every n , m ∈ N , we derive min–max polynomials for P of the form P ( x ) = P n ( x ′ ) x d m , with x ′ = ( x 1 , … , x d − 1 ) , where P n ( x ′ ) is the product of homogeneous harmonic polynomials in two variables. In particular, for every m ∈ N , min–max polynomials for the monomials x 1 … x d − 1 x d m are obtained. Furthermore, we give min–max polynomials for the case where P n ( x ′ ) = ‖ x ′ ‖ n T n ( 〈 a ′ , x ′ 〉 / ‖ x ′ ‖ ) , a ′ = ( a 1 , … , a d − 1 ) ∈ R d − 1 , ‖ a ′ ‖ = 1 , and T n denotes the Chebyshev polynomial of the first kind.