Re: Fourier transform and the like
- From: israel@xxxxxxxxxxx (Robert Israel)
- Date: 11 Oct 2005 23:18:45 GMT
In article <1129058622.804690.94460@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
S. Gammelmark <gammelmark@xxxxxxxxx> wrote:
>I'm well aware, that e^(kx) does not form an orthonormal/orthogornal
>basis with respect to the same inner product as is used in the fourier
>transform/series, but it can, however, be constructed using the
>Gram-Schmidt process.
I'm not sure what "it" is here, but if you're talking about the
real line here in the context of Fourier transform, e^{kx} is not
square-integrable, so Gram-Schmidt can't even get started.
Another way of thinking of Fourier transform is as the
application of the Spectral Theorem to the self-adjoint operator
-i d/dx on L^2(R). This operator is rather special because it
generates translations. Other self-adjoint operators with
continuous spectrum would produce other transforms. Self-adjoint
operators with discrete spectrum lead to orthonormal bases,
analogous to Fourier series. If you want a nice orthogonal basis
for L^2(R), you might try Hermite functions
u_n(x) = pi^(-1/4) 2^(-n/2) (n!)^(-1/2) exp(-x^2/2) H_n(x)
(H_n the Hermite polynomials), which are eigenfunctions for
the harmonic oscillator.
Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.
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