Abraham Avi Arad Hochman

From: orjanjo@math.ntnu.no (Orjan Johansen)
To:Avi Arad
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>,
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