More epsilonized bounds of the Boas-Kac-Lukosz type

In this paper we give a further investigation of the method introduced by the author in [1, Frequency-domain bounds for nonnegative unsharply band-limited functions] for proving bounds for functions with nonnegative Fourier transforms. We also dealt with the question of how large the supremum K^S of all numbers |f(u)| is with f the Fourier transform of a nonnegative integrable function F and f(0) = 1, |f(ku)| ≤ ε for k ∈ S. Here u > 0 and S ⊂ {2, 3, . . .}. This problem was related in [1] to finding the infimum M^S of all numbers M_h = max_ϑ [(1−h(ϑ))/(1− cos ϑ)] over all 2π-periodic even, smooth functions h whose Fourier cosine coefficients a_k vanish for k ∉ S, and it was proved and announced for several cases that M^S (1−K^S ) = 1. In this paper we prove the results announced in [1]. To that end we generalize the method given in [1] to include Fourier transforms f of probability measures on R and a certain generalized function h, and we show that the numbers K^S, M^S are assumed as |f(u)|, M_h for certain allowed f,h. Moreover, we establish a fundamental relation between finding the numbers K^S, M^S and the numbers K^T, M^T where T = {2, 3, . . .}\S. In particular, we show that M^T = 2K^S (2K^S − 1)−1,K^T = 1/2 M^S(M^S − 1)−1 and that M^T (1 − K^T) = 1,K^SK^T = 1/2 , whenever M^S (1 − K^S) = 1.