Reconstruction of characteristic functions of quadratic functionals on trajectories of Gaussian stochastic processes

In this paper, we examine the characteristic functions $Q_J(-i\lambda )$, $\lambda \in R$, of stochastic variables determined by the values of the quadratic functionals $J\left[\tilde{x}\right(t\left)\right]$ on the space $L_2\left[\mathrm{0,}T\right]$ of trajectories of homogeneous Gaussian stochastic processes. We justify a method for calculating such characteristic functions, called reconstruction in the work, the application of which is not related to the use of the well-known Karhunen–Loeve–Pugachev method.