Some extremal properties of positive trigonometric polynomials

A class P_n of even positive trigonometric polynomials tn(?)=a₀ + a₁ cos ?+ ... + a_n cos · n?, satisfying the conditions: a_k ≥0 (k = 0,1, ..., n), a₀ < a₁ is considered. The behavior of the sequence of functionals $$v_n = _{t_n \mathop { \in P_n }\limits^{\inf } } \frac{{t_n \left( 0 \right) - a_o }}{{\left( {\sqrt {a_1 } - \sqrt {a_o } } \right)}}$$ , is studied; two-sided estimations are given for V_n and \(V_\infty = \mathop {\lim }\limits_{n \to \infty } V_n \) .