Nikol'skii inequality for algebraic polynomials on a multidimensional Euclidean sphere
Let \(\mathcal P_{n,m}\) be the set of algebraic polynomials of degree at most \(n\) in \(m\) variables with real coefficients. In this contribution the authors analyze the following Nikol'skii type inequality \[ || P_n||_{C(Sn-1 ) } \leq C(n, m)_q ||P_n||_{L_q (\mathbb S^{n-1} )}, \quad P_n \in \mathcal P_{n,m}, \] with the least constant \(C(n, m)_q\) between the uniform norm \[ ||P_n||_ {C(\mathbb S^{n-1} )} = \max\{|P_n (x)| : x \in \mathbb S^{n-1} \} \] and the \(L_q\) norm \[ ||P_n||_ {L_q(\mathbb S^{n-1} )} = \left(\int_{\mathbb S^{n-1}} |P_n (x)|^q\right)^{1/q}, 1 \leq < \infty, \] of polynomials of a given degree \(n \geq 0\) on the unit sphere. The authors show that the above multidimensional inequality can be reduced to two extremal problems for algebraic polynomials in one variable (see Theorem 2). Also, the authors discuss the uniqueness of an extremal polynomial associated with one of these extremal problems for algebraic polynomials in one variable (cf. Theorem 1).
polynomials that deviate least from zero
scientific article; zbMATH DE number 6381060