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

Abstract

Let Π d n+m−1 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 d n+m−1 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 m d , 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 m d 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.