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   -> Volume 2, Issue 6

Abstract: Wavelet Electrodynamics
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Gerald Kaiser, University of Massachusetts at Lowell

PostPosted: Fri Nov 29, 2002 2:30 pm    
Subject: Abstract: Wavelet Electrodynamics
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Abstract: Wavelet Electrodynamics

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centerline{(To appear in Proc. of {sl Wavelets and Applications/}
conference, Toulouse, 1992)}

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centerline{Gerald Kaiser}
centerline{Department of Mathematics}
centerline{University of Massachusetts at Lowell}
centerline{Lowell, MA 01854, USA}
centerline{e--mail: kaiserg@}

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centerline{f ABSTRACT}
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oindent A wavelet representation of electromagnetic waves (solutions of
Maxwell's equations) is constructed as follows: (a) Using the previously
defined {sl Analytic--Signal transform,/} solutions are extended
from real spacetime $v R^4$ to analytic vector fields on a double tube
domain ${cal T}subset v C^4$ in complex spacetime. (b) The evaluation
maps $e_z , (zin {cal T})$ on the space of such analytic solutions are {sl
conformal wavelets./} They can all be obtained by applying conformal
transformations of spacetime to a single ``basic" wavelet. The basic wavelet,
in turn, is {sl uniquely determined/} by analyticity considerations. (c)
The eight real parameters $zin {cal T}$ have a complete geometric and
physical interpretation: They amount to specifying a location $v x$, a time
$t$, a scale $s$ and a velocity $v v$ for the wavelet $e_z$. When
$v v =v 0$, $e_z$ is a spherical wave which implodes toward
$v x$, builds up to a ball of radius $sqrt{3},|s|$ at time $t$, then
explodes away from $v x$. Wavelets with $v v
e v 0$ are {sl
Doppler--shifted/} versions of the above. (d) An arbitrary solution can be
decomposed into either stationary or moving wavelets. This leads to a
representation of solutions which is analogous to their Fourier
representation, but with the plane waves replaced by localized wavelets.
Such representations may be useful in the efficient analysis of radiation
emitted by moving localized sources. There also exist decompositions
suitable for radiation due to accelerating sources. The wavelet
representation of electromagnetic fields is closely related to the relativistic
coherent--state representations for Klein--Gordon and Dirac fields
developed in earlier work.
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