Re: Fourier transform question
- From: rusty <mr.rusty@xxxxxxx>
- Date: Wed, 01 Aug 2007 11:19:31 +0200
eugene wrote:
While solving a problem i came up with the following question: Let f
be in L^1(R), such that F(f) = a* F for some non-zero complex constant
a. What can be said about a ? * Here F is the Fourier transform.
It is clear that since F^4 = id, we have that a^4 = 1. Is it possible
to have more conditions on a ?
The nth Hermite function is an eigenfunction of the Fourier transform
with eigenvalue (-i)^n and lies in S(R), so every L^p space.
(This is because exp -x^2/2 is an eigenfunction of the FT with
eigenvalue 1 and the "creation operator" A = -d/dx + x
is an eigenoperator for the conjugation action of the
FT with eigenvalue -i. The nth Hermite function is obtained
by applying A^n to exp -x^2/2 and normalising.)
--
rusty
.
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