An estimate of the constant term of a nonnegative trigonometric polynomial with integer coefficients (Q1358528)

A sequence \(\Lambda= \{\lambda_n\}^\infty_{n=1}\) of positive numbers is said to be admissible if \[ \sum^\infty_{n=1} {1\over \lambda_n} <\infty \quad \text{and} \quad \sum^\infty_{n=1} \sin\left({x \over\lambda_n} \right)\geq 0 \] for all \(x\geq 0\). If, moreover, \[ \sum^\infty_{n=1} \sin\left( {x\over\lambda_n} \right)>0 \] for all \(x>0\), then the sequence \(\Lambda\) is said to be strictly admissible. This definition was introduced by the first author. The aim of the present paper is to study admissible sequences and to use them to estimate the constant term of a nonnegative trigonometric polynomial with integer coefficients.