[Cst-1b] Fourier Transform product

[M.Y.W.Y.B.]

Dear Phebe, 

it's no wonder that you had difficulties with showing that the product of
two GWs is another single GW because it isn't, unless you define the
functional form of GWs differently, ie with an extra real factor in front of
the formula. See my notes under (2) for more details. This mistake affects
questions 8.4 and 8.6 of the problem sheet he handed out (corresponding to
some Tripos questions).

IMHO there is another mistake in the notes and in problem 6.4 where he
claims that the GW "unifies" the Fourier and the space domain. Again, this
can be overcome by adding an extra factor of something like 1/Sqrt(2*PI*a)
(I believe). So I suggest a functional form for the GW of 
f(x) = 1/Sqrt(2*PI*a) exp(-imx) exp-(x-x0)^2/a^2

See notes under (1).

Moritz

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(1)
In the notes and in the answer to problem 6.4 it says:
   "...in the limit our expansion basis becomes [...] exp(-imx), 
   the ordinary Fourier basis". 
This is correct since 
f(x) = integral(F(m) exp(-imx) dm) between -inf ... +inf

But then he claims that if a->0 the limit becomes delta(x-x0) which is the
   "expansion basis implementing pure space-domain sampling". 
While it's true that the Dirac delta function is the expansion basis for the
space domain
since 
f(x) = integral(f(z) delta(x-z) dz) between -inf ... +inf
it's wrong that lim(f(x)) = Dirac-delta(x-x0) for a->0, m->0. In fact the
limit is the *Kronecker* delta function which is surely NOT the expansion
basis for the time domain.



(2)
In problem 8.4 and 8.6 he claims that the product of two Gabor wavelets (GW)
is just another Gabor wavelet because "the arguments of the exponentials
simply add within each exponential". That is wrong because the exponents of
the Gaussian part of the GW do *not* simply add in general, ie if you don't
happen to have the same exponent in both arguments of the product. You
essentially have to prove that the product of the two Gaussian exponentials
is another Gaussian exponential which is not easy. In fact it only works if
you put an *extra factor* as parameter in front of the exponential.
Therefore
the product of two GW has *not* got the same functional form as stated in
the notes and thus the set of all GW is *not* closed under convolution. It
would work, however, if the GW was defined as
f(x) = A exp(-imx) exp-(x-x0)^2/a^2