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

We discuss three related extremal problems on the set $ \mathbb{S}^{m - 1} $ of Euclidean space ? ^m of dimension m ≥ 2. (1) The norm of the functional F(h) = FhP _n = ∫_?(h) P _n (x)dx, which is equal to the integral over the spherical cap ?(h) of angular radius arccos h, ?1 < h < 1, on the set $ \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 ?(h) by the subspace $ \mathbb{S}^{m - 1} $ ) that are orthogonal to the space of polynomials $ \mathbb{S}^{m - 1} $ ) of the function χ _h by the space of polynomials ₁ ^? on the interval (?1, 1) with ultraspheric weight ?(t) = (1 ? t ²) ^α , α = (m ? 3)/2.