Eigenwavelets of the Wave equation

Physical wavelets are localized solutions of the scalar wave equation or Maxwell’s equations, obtained by extending fundamental solutions to complex space-time in the sense of hyperfunctions. The imaginary space-time variables y, which must form a time-like vector, act as scale parameters generalizing the scale variable of wavelets in one dimension. They determine the shape of the wavelets in space-time, making them pulsed beams that can be focused as tightly as desired around a single ray by letting y approach the light cone. Furthermore, the absence of any sidelobes makes them especially attractive for communications, remote sensing and other applications using acoustic or electromagnetic waves. I review the basic ideas in the scalar case in R, then compute sources whose realization should make it possible to emit and absorb such wavelets.