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

In this paper, we examine the characteristic functions $Q_J(-i\lambda)$, $\lambda \in {\mathbb R}$, of stochastic variables determined by the values of the quadratic functionals $\mathsf{J}[\tilde{x}(t)]$ on the space ${\mathbb L}_2 [0, T]$ 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.