The Inverse Source Problem for Wavelet Fields

Abstract

The theory of the inverse source problem is employed to compute a class of continuously distributed and compactly supported three-dimensional (volume) sources that radiate the scalar wavelets investigated by Kaiser as well as certain electromagnetic generalizations of these scalar fields. These efforts have shown that the scalar wavelet fields can be radiated by a distributional source (generalized function) supported on a circular disk of radius a or an oblate spheroid surrounding that disk. Our main goal here is to replace this distributional source by a more conventional volume source that radiates the same wavelet field outside its support volume. The equivalent volume sources computed in this paper are supported on (three-dimensional) spherical shells whose outer radius a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sub> > a and inner radius a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sub> < <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sub> are arbitrary. These sources are analytic functions of position within their support volumes for any finite, but arbitrarily large temporal frequency omega, and possess minimum L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> norm among all possible solutions to the inverse source problem with the given support volume constraint. Electromagnetic versions of the wavelet sources and fields are shown to be easily derived from their scalar wave counterparts.