An explicit class of min-max polynomials on the ball and on the sphere (Q547877)

Let \(\Pi^d_k\) be the set of polynomials in \(d\) variables of total degree no greater than \(k\), with real coefficients. Given a homogeneous polynomial \(\mathcal{P}\in \Pi^d_{n+m} \), of degree \(n+m\), this paper deals with the problem of finding \(p^\ast\in\Pi^d_{n+m-1}\) such that \((\mathcal{P}-p^\ast)w\) has least max norm on the unit ball \(B^d\) and the unit sphere \(S^{d-1}\) in dimension \(d\), \(d\geq 2\), where \(w\) is a weight function. Min-max polynomials \(\mathcal{P}-p^\ast\) are given for several types of polynomials \(\mathcal{P}\) and the corresponding minimum deviation is calculated. One case is that in which \(\mathcal{P}(x)=p_n(x^\prime)x_d^m\), where \(p_n\) is the product of homogeneous harmonic polynomials in two variables and \(x^\prime=(x_1,\dots ,x_{d-1})\). It is also analyzed the case \(p_n(x^\prime)=\|x^\prime\|^n T_n(\langle a^\prime,x^\prime \rangle/\|x^\prime\|)\), where \(a^\prime=(a_1,\dots ,a_{d-1})\), \(\|a^\prime\|=1\), and \(T_n\) is the Chebyshev polynomial of the first kind.