Several extremal approximation problems for the characteristic function of a spherical layer

We discuss three interrelated extremal problems on the set P n,m of algebraic polynomials of a given degree n on the unit sphere S m-1 of the Euclidean space ℝ m of dimension m ≥ 2. (1) Find the norm of the functional F(η) = F hP n = ∫ G(η)P n(x)dx, which is the integral over the spherical layer G(η) = {x = (x 1,...,x m) ∈ S m-1: h′ ≤ x m ≤ h″} defined by a pair of real numbers η = (h′, h″), -1 ≤ h′ < h″ ≤ 1, on the set P n,m with the norm of the space L(S m-1) of functions summable on the sphere. (2) Find the best approximation in L ∞(S m-1) of the characteristic function χ η of the layer G(η) by the subspace P n,m ⊥ of functions from L ∞(S m-1) that are orthogonal to the space of polynomials P n,m. (3) Find the best approximation in the space L(S m-1) of the function χ η by the space of polynomials P n,m. We present a solution of all three problems for the values h′ and h″ that are neighboring roots of the polynomial in a single variable of degree n + 1 that deviates least from zero in the space L 1 φ{symbol} (-1, 1) of functions summable on the interval (-1, 1) with ultraspherical weight φ{symbol}(t) = (1 - t 2) α, α = (m - 3)/2. We study the respective one-dimensional problems in the space of functions summable on (-1, 1) with an arbitrary not necessarily ultraspherical weight. © 2012 Pleiades Publishing, Ltd