The Taikov Functional in the Space of Algebraic Polynomials on the Multidimensional Euclidean Sphere

We discuss three related extremal problems on the set $\mathscr P_{n,m}$ of algebraic polynomials of given degree $n$ on the unit sphere $\mathbb S^{m-1}$ of Euclidean space $\mathbb R^m$ of dimension $m\ge 2$. (1) The norm of the functional $F(h)=F_hP_n=\int_{\mathbb C(h)}P_n(x)\,dx$, which is equal to the integral over the spherical cap $\mathbb C(h)$ of angular radius $\operatorname{arccos} h$, $-1<h<1$, on the set $\mathscr P_{n,m}$ with the norm of the space $L(\mathbb S^{m-1})$ of summable functions on the sphere. (2) The best approximation in $L_\infty(\mathbb S^{m-1})$ of the characteristic function $\chi_h$ of the cap $\mathbb C(h)$ by the subspace $\mathscr P^\bot_{n,m}$ of functions from $L_\infty(\mathbb S^{m-1})$ that are orthogonal to the space of polynomials $\mathscr P_{n,m}$. (3) The best approximation in the space $L(\mathbb S^{m-1})$ of the function $\chi_h$ by the space of polynomials $\mathscr P_{n,m}$. 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 $L_1^\phi$ on the interval $(-1,1)$ with ultraspheric weight $\phi(t)=(1-t^2)^\alpha$, $\alpha=(m-3)/2$.