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

We discuss three related extremal problems on the set $\mathcal P_{n,m}$ of algebraic polynomials of a given degree $n$ on the unit sphere $\mathbb S^{m-1}$ of the Euclidean space $\mathbb R^m$ of dimension $m\ge2$. (1) The norm of the functional $F(\eta)=F_hP_n=\int_{\mathbb G(\eta)}P_n(x)dx$, which is equal to the integral over the spherical layer $\mathbb G(\eta)=\{x=(x_1,\dots,x_m)\in\mathbb S^{m-1}\colon h'\le x_m\le h''\}$ defined by a pair of real numbers $\eta=(h',h'')$, $-1\le h'<h''\le1$, on the set $\mathcal P_{n,m}$ with the norm of the space $L(\mathbb S^{m-1})$ of functions summable on the sphere. (2) The best approximation in $L_\infty(\mathbb S^{m-1})$ of the characteristic function $\chi_\eta$ of the layer $\mathbb G(\eta)$ by the subspace $\mathcal P^\bot_{n,m}$ of functions from $L_\infty(\mathbb S^{m-1})$ that are orthogonal to the space of polynomials $\mathcal P_{n,m}$. (3) The best approximation in the space $L(\mathbb S^{m-1})$ of the function $\chi_\eta$ by the space of polynomials $\mathcal P_{n,m}$. We present the solution of all three problems for the values $h'$ and $h''$ which are neighboring roots of the polynomial in a single variable of degree $n+1$ that deviates the least from zero in the space $L_1^\phi(-1,1)$ on the interval $(-1,1)$ with ultraspherical weight $ \phi(t)=(1-t^2)^\alpha$, $\alpha=(m-3)/2$. We study the respective one-dimensional problems in the space of functions summable on $(-1,1)$ with arbitrary not necessary ultraspherical weight.