Wavelet analysis for the solutions of the wave equation

Abstract

The new approach to the wavelet analysis for the solutions of the homogeneous wave equation in three spatial dimensions is presented. The approach is based on the ideas suggested by G. Kaiser but has different implementation and has some advantages versus the known approach. A new physical wavelet for this wavelet analysis is also presented, with the brief discussion of its main properties. The wavelet analysis has become widely used during the last twenty years and it has a lot of applications nowadays. However most of them are in the field of the numerical processing of the experimental data, digital images, astronomical, geophysical and medical data and other applications of that kind (see, for example, [1], [2]). The amount of the results in the application of the methods of the continuous wavelet analysis to the solutions of the differential equations is not large. In particular the continuous wavelet analysis for the solutions of the three-dimensional homogeneous wave equations with a constant wave speed was first developed by G. Kaiser in his book [3]. He suggests a method for decomposition of the solutions of the wave equation in terms of the localized solutions of the same equation based on the analytic signal transform and on the theory of the analytic functions of several variables. The wavelet for such decomposition was also suggested, and the class of such wavelets was named ’physical wavelets’. The sort of the wavelet analysis developed by Kaiser is close to the holomorphic wavelet transform (see, for example, [2]). However, this approach may be found unfamiliar by the people who deal with the wavelet analysis within the framework of the signal and image processing. The aims of this paper are as follows. First we develop the wavelet analysis for the solutions of the wave equation not involving the analytic signal transform. The ideas, which we base on, were suggested by Kaiser in [3], however their implementation here differs from his approach. The method we use is intrinsic to the common continuous wavelet transform and we hope will be more familiar to the people who work in the area of the signal and image processing. Our approach also provides some advantages in comparison to that, suggested by Kaiser. We enlarge the class of solutions which can be used as the mother wavelets for the analysis. The second aim is to find a new physical wavelet for our method, i.e., to find the solution of the wave equation which will be an admissible wavelet. The new wavelet is constructed by means of the field of point sources and of proxy wavelets using the technique suggested by G. Kaiser. This new spherically symmetric physical wavelet has good properties such as exponential localization in both the coordinate and the Fourier