From: orjanjo@math.ntnu.no (Orjan Johansen)
Newsgroups: sci.math
Subject: Re: Distribution theory, Fourier transforms, and Parsevals theorem
Date: 7 Oct 1998 14:17:12 GMT

In article <6vc4h3$4hl$1@nnrp1.dejanews.com>,
 <jhunpingco@my-dejanews.com> wrote:
>
>I have  questioned regarding distribution theory,  Fourier transforms,
>and  Parsevals  theorem.  Usually,  there are  two  versions  of  this
>theorem: one  version for periodic  signals and the other  version for
>aperiodic signals.  For example, given  x(t) = x(t+T), one  version of
>the theorem is
>
>1/T int_{-T/2}^{T/2} abs(x(t))^2 dt = sum_k abs(a_k)^2
>
>where a_k are the Fourier coefficients. The other version for aperiodic
>signals is
>
>int_{-infinity}^{infinity} abs(x(t))^2 dt = int_{-infinity}^{infinity}
>abs(X(u)) ^2 dt
>
>where  X(u)  is  the  Fourier  transform of  x(t).  I'm  wondering  if
>distribution  theory  unifies  these two  formulations.

I don't know about distribution theory, but one theory which does
unify these is the theory of harmonic analysis on (locally compact
abelian) groups.  The relevant theorem there is:

Plancherel's theorem
--------------------
The Fourier transform from L^2(G) to L^2(G^), where G^ is the dual
group of G, is an isometry.

The integrals and sums above all calculate the square of the L^2-norm.
In the first case the group is the multiplicative unit circle of
complex numbers and its dual is the integers;  In the second case the
group is the additive real numbers, which is self-dual.

Greetings,
Ørjan.

-- 
'What Einstein called "the happiest thought of my life" was his
realization that gravity and acceleration are both made of orange
Jello.'  - from a non-crackpot sci.physics.relativity posting