Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes

[摘要] With the notation K := R (mod 2pi), \\p\\(Llambda(K)) := (integral(K)\p(t)\(lambda)dt)(1/lambda) and M-lambda(p) := ((1)/(2pi) integral(K)\p(t)\(lambda)dt)(1/lambda) we prove the following result. Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with real coefficients that satisfies \\p\\(L2(K)) less than or equal to An(1/2) and \\p'\\(L2(K)) greater than or equal to Bn-3/2. Then M-4(p) - M-2(p) greater than or equal to epsilonM(2)(p) with epsilon := ((1)/(111))((B)/(A))(12). We also prove that M-infinity(1 + 2p) - M-2(1 + 2p) greater than or equal to (root4/3 - 1) M-2 (1 + 2p) and M-2(p) - M-1(p) greater than or equal to 10(-31) M-2(p) for every p is an element of A(n), where A(n) denotes the collection of all trigonometric polynomials of the form p(t) := p(n)(t) := (n)Sigma(j=1) a(j) cos(jt + x(j)), a(j) = +/- 1, alpha(j) is an element of R. (C) 2003 Elsevier Inc. All rights reserved.

[发布日期] 2003-12-01

[关键词] trigonometric polynomials;merit factor;unimodular trigonometric polynomials;Littlewood class