Geodesic
On the Representation of Localized Functions in~$\mathbb R^2$ by Maslov's Canonical Operator
Matematičeskie zametki, Tome 96 (2014) no. 1, pp. 88-100

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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.
Keywords: wave equation, asymptotics, localized initial data, integral representation, invariant Lagrangian manifold, Maslov's canonical operator. 
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     author = {V. E. Nazaikinskii},
     title = {On the {Representation} of {Localized} {Functions} in~$\mathbb R^2$ by {Maslov's} {Canonical} {Operator}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {88--100},
     publisher = {mathdoc},
     volume = {96},
     number = {1},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_96_1_a7/}
}
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V. E. Nazaikinskii. On the Representation of Localized Functions in~$\mathbb R^2$ by Maslov's Canonical Operator. Matematičeskie zametki, Tome 96 (2014) no. 1, pp. 88-100. http://geodesic.mathdoc.fr/item/MZM_2014_96_1_a7/