The Taikov functional in the space of algebraic polynomials on the multidimensional Euclidean sphere

Abstract

We discuss three related extremal problems on the set

of algebraic polynomials of given degree n on the unit sphere $\mathbb{S}^{m - 1}$ of Euclidean space ℝ^m of dimension m ≥ 2. (1) The norm of the functional F(h) = FhP _n = ∫_ℂ(h) P _n (x)dx, which is equal to the integral over the spherical cap ℂ(h) of angular radius arccos h, −1 < h < 1, on the set

with the norm of the space L($\mathbb{S}^{m - 1}$) of summable functions on the sphere. (2) The best approximation in L _∞($\mathbb{S}^{m - 1}$) of the characteristic function χ _h of the cap ℂ(h) by the subspace

of functions from L _∞($\mathbb{S}^{m - 1}$) that are orthogonal to the space of polynomials

. (3) The best approximation in the space L($\mathbb{S}^{m - 1}$) of the function χ _h by the space of polynomials

. We present the solution of all three problems for the value h = t(n,m) which is the largest root of the polynomial in a single variable of degree n + 1 least deviating from zero in the space L ^ϕ₁ on the interval (−1, 1) with ultraspheric weight ϕ(t) = (1 − t ²)^α, α = (m − 3)/2.