The Bernstein class \(B\) consists of entire functions of exponential type \(\leq \pi\) that are bounded on the real axis. It is well known that if \(F\in B\) is a Fourier-Stieltjes transform, then the corresponding measure is necessarily concentrated on the torus \(\mathbb{T}\). In the present paper, certain subclasses of \(B\) are considered, while functions are constructed that belong to these subclasses but are not Fourier-Stieltjes transforms. Particular attention is paid to the distribution of zeros of such functions. The results obtained are applied to determine whether the completeness of a system \(\exp(i\lambda_nt)\) of exponentials in the spaces \(C(\mathbb{T})\) and \(L^p(\mathbb{T})\) is stable under small perturbations of the \(\lambda_n\).