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Poincaré wavelets and their applications to Gaussian beam summation

The decomposition of solutions of an initial-boundary value problem for the wave equation in terms of localized solutions is presented. The decomposition is constructed with the mathematical technique of the continuous affine Poincaré analysis. In the case of two spatial dimensions, the obtained results are given as a fourfold integrals in the space of parameters of localized solutions. We apply this formula to the Helmholtz equation. As an example, we obtain a decomposition of the Green's function for the Helmholtz equation in terms of Gaussian beams. The results are in good agreement with a known formula of the Gaussian beam summation due to Popov.