Geodesic
Solution of the exponential moment problem in the space $L^2(0,\infty)$
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 193-194

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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. 
@article{ZNSL_1977_73_a12,
     author = {S. A. Avdonin},
     title = {Solution of the exponential moment problem in the space $L^2(0,\infty)$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {193--194},
     publisher = {mathdoc},
     volume = {73},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1977_73_a12/}
}
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S. A. Avdonin. Solution of the exponential moment problem in the space $L^2(0,\infty)$. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 193-194. http://geodesic.mathdoc.fr/item/ZNSL_1977_73_a12/