Statistical aspects of acausal pulses in physics and wavelets applications

Abstract

In the mathematical theory of distributions are widely used test-functions (which differ to zero only on a limited interval and have continuous derivatives of any order on the whole real axis). The use of such functions is also recommended in Fourier analysis of wavelets. However, less attention was given to connections between test-functions and equations used in mathematical physics (as wave equation). This paper shows that test-functions, considered at the macroscopic scale (that means not as δ -functions) can represent solutions for the wave-equation, under the form of acausal pulses (which appear under initial null conditions and without any source-term to exist). This implies the necessity for some supplementary requirements to be added to the wave-equation, so as the possibility of appearing such pulses to be rejected. It will be shown that such a possibility represents in fact a kind of bifurcation point, and a statistic interpretation (based on probability for state-variables to make certain jumps) is presented for justifying the fact that such pulses are not observed. Finally the advantage of using practical test function for wavelets processing is presented. © Springer-Verlag Berlin Heidelberg 2005.