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   -> Volume 2, Issue 9


Preprint available: Wavelet Electrodynamics, Part II
 
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Gerald Kaiser, University of Massachusetts at Lowell.
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PostPosted: Fri Nov 29, 2002 3:31 pm    
Subject: Preprint available: Wavelet Electrodynamics, Part II
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Preprint available: Wavelet Electrodynamics, Part II

*** this is a tex-file, cut here ***
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defv#1{{f#1}}
defc#1{{cal#1}}def #1{ ilde#1}


centerline{f WAVELET ELECTRODYNAMICS, Part II:}


vglue 1ex

centerline{f Electromagnetic Wavelets with Polarization}

vglue 3ex


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

vglue 2ex

centerline{June 2, 1993}

vglue 2ex
centerline{f ABSTRACT}

vglue 1ex

oindent The representation of free
electromagnetic waves (solutions of Maxwell's equations) as
superpositions of scalar wavelets with vector coefficients
developed earlier is generalized to wavelets with
polarization. The construction is canonical and proceeds in four
stages: {f(1)} The {sl analytic--signal transform} is used to
extend fields from real spacetime to a tube domain $c T$ in
complex spacetime. {f(2)} The {sl evaluation maps,/} which
send any field $v F$ to the values $ v F(z)$ of its extension
at points $zinc T$, are bounded linear maps on the space of
solutions. Their adjoints $v W_z$ are the
electromagnetic wavelets. These are matrix--valued to include all
allowed states of polarization. {f(3)} The eight real parameters
$z=x+iyin c T$ are given a complete physical interpretation:
$x=(v x, t)inv R^4$ is interpreted as a spacetime point about
which $v W_z$ is {sl focussed,/} i.e., $v W_{x+iy}$ is localized
around the space point $v x$ at time $t$. The imaginary
spacetime variable $y=(v y, s)$ is timelike. Its
Lorentz--invariant magnitude $lambda=(s^2-|v y|^2)^{1/2}$ is
interpreted as the {sl scale} of the wavelet (i.e., its width at the
time of maximal focus), and its direction $v v=v y/s$ as the {sl
velocity} of its center. In particular, wavelets $v W_z$
parameterized by {sl Euclidean/} points $z=(v x, is)$ (real space
and imaginary time) have stationary centers; the real space point
$v x$ is their point of localization, and the imaginary time is
their scale. The set $Esubsetc T$ of all such points is accordingly
called {sl wavelet space.} {f(4)} A resolution of unity is
derived for the Hilbert space of solutions which gives a
representation of any solution $v Fin c H$ as a superposition of
the wavelets $v W_z$ parameterized by $zin E$.
ye
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