Some extremal problems for trigonometric polynomials

Three extremal problems for trigonometric polynomials are studied in this paper. The first was initiated by Maiorov. It relates to the trigonometric polynomials with n nonzero harmonics. The L-p-norm of the Weyl derivative is compared with the L-q-norm of the polynomial. The other two problems have appeared in the recent paper by Oswald. They deal with the polynomials of degree n. The minimum of L-p-norm with respect to the choice of phases is compared with l(q)-norm of its coefficients. (C) 2003 Elsevier Inc. All rights reserved.