Abstract
We discuss three related extremal problems on the set Open image in new window of algebraic polynomials of given degree n on the unit sphere \( \mathbb{S}^{m - 1} \) of Euclidean space ℝm of dimension m ≥ 2. (1) The norm of the functional F(h) = FhPn = ∫ℂ(h)Pn(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 Open image in new window 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 χh of the cap ℂ(h) by the subspace Open image in new window of functions from L∞(\( \mathbb{S}^{m - 1} \)) that are orthogonal to the space of polynomials Open image in new window . (3) The best approximation in the space L(\( \mathbb{S}^{m - 1} \)) of the function χh by the space of polynomials Open image in new window . 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 L1ϕ on the interval (−1, 1) with ultraspheric weight ϕ(t) = (1 − t2)α, α = (m − 3)/2.
Published Version
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