Let \(f\) be an analytic function of bounded characteristic in the unit disc \(\mathbb{D}\). Assume that the boundary values of \(f\) on the circumference \(\mathbb{T}\) belong to \(L^2 (\mathbb{T})\). Denote by \(\widehat f(n)\), \(n \in \mathbb{Z}\), the Fourier coefficients of \(f\). The author studies the problem of characterization of the increasing sequences \(\{\alpha_n\}_1^\infty\) such that the following implication \[ \bigl |\widehat f(-n) \bigr |\leq 1/ \alpha_n,\;n \in \mathbb{N}, \Rightarrow \quad \widehat f(-n) = 0,\;n \in \mathbb{N}, \tag{1} \] holds. The results of the paper solve the problem under some additional restrictions on \(\{\alpha_n\}^\infty_1\). In particular, the author proves that, if \(\{\alpha_n\}^\infty_1\) is logarithmically convex, then (1) holds if and only if \[ \sum^\infty_1 (\log \alpha_n)/(1 + n^{3/2}) = + \infty. \] Applications of the results to the study of the functions of bounded characteristic with infinitely differentiable boundary values are given.