On the Representation of Localized Functions in $\mathbb R^2$ by Maslov's Canonical Operator

We prove that localized functions can be represented in the form of an integral over a parameter, the integrand being Maslov's canonical operator applied to an amplitude obtained from the Fourier transform of the function to be represented. This representation generalizes an earlier one obtained by Dobrokhotov, Tirozzi, and Shafarevich and permits representing localized initial data for wave type equations with the use of an invariant Lagrangian manifold, which simplifies the asymptotic solution formulas dramatically in many cases.