Some trigonometric extremal functions and the Erdos-Turan type inequalities

In this paper we study a type of trigonometric extremal problem which originates in the work of Arne Beurling in the late 1930s and concerns approximations to the function sgn(x). The extremal problem is formulated and solved within the theory of Jacobi's polynomials and the theory of reproducing kernel Hilbert spaces of polynomials (cf. Li [9]). As an application of our results, an Erdos-Turan type inequality is obtained for spherical caps. Further discussion of the history of such problems is given by Selberg [12].