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

Let Πn+m−1d denote the set of polynomials in dd variables of total degree less than or equal to n+m−1n+m−1 with real coefficients and let P(x)P(x),x=(x1,…,xd)x=(x1,…,xd), be a given homogeneous polynomial of degree n+mn+m in dd variables with real coefficients. We look for a polynomial p∗∈Πn+m−1d such that P−p∗P−p∗ has least max norm on the unit ball and the unit sphere in dimension dd,d≥2d≥2, and call P−p∗P−p∗ a min–max polynomial. For every n,m∈Nn,m∈N, we derive min–max polynomials for PP of the form P(x)=Pn(x′)xdm, with x′=(x1,…,xd−1)x′=(x1,…,xd−1), where Pn(x′)Pn(x′) is the product of homogeneous harmonic polynomials in two variables. In particular, for every m∈Nm∈N, min–max polynomials for the monomials x1…xd−1xdm are obtained. Furthermore, we give min–max polynomials for the case where Pn(x′)=‖x′‖nTn(〈a′,x′〉/‖x′‖)Pn(x′)=‖x′‖nTn(〈a′,x′〉/‖x′‖),a′=(a1,…,ad−1)∈Rd−1a′=(a1,…,ad−1)∈Rd−1,‖a′‖=1‖a′‖=1, and TnTn denotes the Chebyshev polynomial of the first kind.