Integral representation of solutions of the wave equation based on Poincaré wavelets

We present here an exact integral representation of solutions of the wave equation with constant coefficients in two spatial dimensions in terms of localized solutions. A solution is given as a superposition of localized solutions each of which lives in the reference system, which moves in the x direction with velocity v. To obtain any solution, we must take into account all |v| ≤ c, where c is the velocity of wave propagation, and use also shifts and scaling. The representation is constructed by means of space-temporal wavelet theory which is applied to the section of a solution in the plane y = 0.