Spectral synthesis for exponentials and logarithmic length (Q2089816)

Let \(\{ v_n \}_{n\in \mathbb N}\) be a complete and minimal system of vectors in a separable Hilbert space \(H\), that is, \(\operatorname{Span}\{ v_n \}^- = H\) and \(\operatorname{Span}\{ v_n \}_{n\neq m}^- \neq H\) for any \(m\). For any such sequence there exists its unique biorthogonal system \(\{ w_n \}_{n\in \mathbb N}\) such that \(\langle v_n, w_m \rangle= \delta_{nm}\). Hereditary completeness can be understood as a weakest form of reconstruction of a vector \(f\) from its generalized Fourier series \(\sum_{n\in \mathbb N} \langle f, w_n\rangle v_n\), since it is equivalent to the fact that each vector \(f \in H\) can be approximated by linear combinations of the partial sums of its Fourier series. The authors are interested in the case when \(H=L^2(-\pi, \pi )\) and \(v_n\) is a system of exponentials, \(v_n= e^{i\lambda_n t}\) for some set \(\Lambda = \{ \lambda_n \}_{n\in \mathbb N} \subset \mathbb C\). They consider the generating function \(G(z) = \lim_{R\to \infty} \prod_{\lambda \in \Lambda, |\lambda |<R} (1- \frac{z}{\lambda} )\) of the system \(\{ e^{i\lambda t} \}_{\lambda \in \Lambda}\) and are interested in the case when the function \(G\) is small outside some lacunary sequence of intervals \(\{ I_k \}_{k=1}^\infty\), where \(I_k=[ \rho_k-d_k, \rho_k + d_k], \, 2\rho_k \le \rho_{k+1}, \, 1 < d_k \le 0.1 \rho_k\). They prove that, under some additional restrictions, the system of exponentials is hereditarily complete if and only if the total logarithmic length of these intervals is infinite, i.e. \(\sum_{k=1}^\infty \frac{d_k}{\rho_k} = \infty\).