NORMS OF DIRICHLET KERNELS AND SOME OTHER TRIGONOMETRIC POLYNOMIALS IN L(P)-SPACES

The following problem is considered. Let a = {a(n)}n=1M = {a(n1), ..., n(m)}n1, ...,n(m)=1M1, .., M(m) be a finite m-fold sequence of nonnegative numbers such that if n greater-than-or-equal-to k then a(n) less-than-or-equal-to a(k), and Q(x) = SIGMA(n=1)M, a(n)e(inx). The purpose of the work is to give best possible upper estimates of the norms \\Q(x)\\p and \\Q(x)\\delta, p with delta > 0 in terms of the coefficients {a(n)}. The Dirichlet kernels D(U)(x) = SIGMA(n is-an-element-of U)e(inx) with U is-an-element-of A1 present a particular case of Q(x). Bibliography: 14 titles.