A criterion for the Bessel property of systems of the form \(\{\exp (i\lambda_n t)\sin (\mu_n t)\}\) and \(\{\exp (i\lambda_n t)\cos (\mu_n t)\}\) (Q5942169)

A system of functions \(\{u_n\}_{n=1}^{\infty}\) is said to have the Bessel property in the space \(L_2(0,\pi)\) if there exists a constant \(\beta>0\) such that \(\sum_{n=1}^{\infty}|(u_n,f)|^2\leq\beta\|f\|^2\) for all \(f\in L_2(0,\pi)\), where \(\|\cdot\|\) is the norm in \(L_2(0,\pi)\) and \((\cdot,\cdot)\) is the inner product in \(L_2(0,\pi)\). The author considers the systems \(\{\varphi_n\}_{n=1}^{\infty}\) and \(\{\psi_n\}_{n=1}^{\infty}\) with \[ \varphi_n=\exp(i\lambda_n t)\sin(\mu_n t), \qquad\psi_n=\exp(i\lambda_n t)\cos(\mu_n t), \] and the system \(\{u_n\}_1^{\infty}\) with \(u_n=c_1\varphi_n+c_2\psi_n\), and \(c_1\) and \(c_2\) are some complex coefficients such that \(c_2\pm i c_1\not=0\). It is supposed that \(\{\mu_n\}_1^{\infty}\) has no finite accumulation points and that \(\{\mu_n\}_1^{\infty}\) and \(\{\lambda_n\}_1^{\infty}\) satisfy the Carleman conditions. Necessary and sufficient conditions are obtained for the system \(\{u_n/\|u_n\|\}_1^{\infty}\) to have the Bessel property.