Solution of the exponential moment problem in the space $L^2(0,\infty)$

We study the system of exponentials $\{\exp(-\lambda_kt)\}\subset L_2(0,+\infty)$, $\lambda_k=c\{1+C(1/k)\}k^\beta$, $\beta>1$, $c>0$, An asymptotic formula is obtained for the biorthogonal system $\theta_k$,,
$$ \theta_k=\exp 2k[v.p.\int_0^\infty\tau^{1/\beta}(\tau^2-1)^{-1}d\tau+0(1)]. $$
is obtained. In the space $L^2(0,\infty)$ we consider the moment problem.