The integral approximation of the characteristic function of a spherical cap by algebraic polynomials

In the space $L(\mathbb S^2)$ on the unit sphere $\mathbb S^2$ of the three-dimensional Euclidean space $\mathbb R^3$, the problem on the best approximation of the characteristic function of a spherical cap by the set of algebraic polynomials of given (total) degree in three real variables with real coefficients is solved.