Some extremal problems of harmonic analysis and approximation theory

The paper is devoted to a survey of the main results obtained in the solution of the Turán and Fejér extremal problems on the torus; the Turán, Delsarte, Bohmann, and Logan extremal problems on the Euclidean space, half-line, and hyperboloid. We also give results obtained when solving a similar problem on the optimal argument in the module of continuity in the sharp Jackson inequality in the space $L^2$ on the Euclidean space and half-line. Most of the results were obtained by the authors of the review. The survey is based on a talk made by V. I. Ivanov at the conference «6th Workshop on Fourier Analysis and Related Fields, Pecs, Hungary, 24-31 August 2017». We solve also the problem of the optimal argument on the hyperboloid. As the basic apparatus for solving extremal problems on the half-line, we use the Gauss and Markov quadrature formulae on the half-line with respect to the zeros of the eigenfunctions of the Sturm–Liouville problem. For multidimensional extremal problems we apply a reduction to one-dimensional problems by means of averaging of admissible functions over the Euclidean sphere. Extremal function is unique in all cases.