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>, 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