New exact integral representations of solutions of the wave equation based on continuous wavelet analysis
Department of Mathematical Physics, Physics Faculty, St. Petersburg University
given at strobl07 (22.06.07 09:30)
Solutions of the wave equation with constant coefficients in a three-dimensional space are presented as superpositions of its localized solutions, which we call physical wavelets, following G.Kaiser. The talk consists of two parts.
The first one is based on my works with M.S.Sidorenko [1-4]. The space of square integrable solutions is decomposed into a direct sum of two subspaces. In each subspace, the formalism of continuous wavelet analysis is developed. The choice of the mother physical wavelet is discussed. A family of wavelets is constructed from this wavelet with the help of transformations of dilation, spatial translation and rotation. Next the wavelet transform of an arbitrary solution from the subspace is defined. An isometry and a reconstruction formula are proved.
The comparison of results with the results of G. Kaiser is given.
The second part contains an integral representation for solutions of the boundary-initial value problem for the wave equation in a half-space. Physical wavelets are constructed by dilation, translation and Lorentz transformations from four mother physical wavelets .
 Perel M.V. and Sidorenko M.S., 2003, Wavelet Analysis in Solving the Cauchy Problem for the Wave Equation in Three-Dimensional Space In:
Mathematical and numerical aspects of wave propagation: Waves 2003, Ed G C Cohen, E Heikkola, P Jolly and P Neittaanmaki (Springer-Verlag) pp 794-798
 Perel M.V. and Sidorenko M.S., 2006, Wavelet analysis for the solution of the wave equation, In: Proc. of the Int. Conf. DAYS on DIFFRACTION 2006, Ed. I V Andronov (SPbU), pp 208-217
 Perel M.V. and Sidorenko M.S., 2007, New physical wavelet 'Gaussian wave packet', Journal of Physics A: Mathematical and Theoretical 40, pp 3441-3461,
 Perel M.V. and Sidorenko M.S., Wavelet-based integral representation for solutions of wave and Klein-Gordon equations, to appear
 Perel M.V. Wavelet-analysis in solving the boundary-initial value problem for the wave equation in a half-space, to appear