Solution of the exponential moment problem in the space L²(0, ∞)

Abstract

We study the system of exponentials$\left\{ {\exp \left( { - \lambda _K t} \right)} \right\} \subset L_ \propto \left( {0, + \infty } \right), \lambda _K = c\left\{ {1 + C\left( {{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle K$}}} \right)} \right\}\kappa ^\beta$, An asymptotic formula is obtained for the biorthogonal system θ_k,

$$\theta _\kappa = \exp 2\kappa \left[ {v.p.\int\limits_0^\infty {\tau ^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle \beta $}}} } \left( {\tau ^2 - 1} \right)^{ - 1} d\tau + 0\left( 1 \right)} \right]$$

is obtained.

In the space L²(0, ∞) we consider the moment problem