Re: Fourier Transform, Smooth Functions

Thank you very much David P, David U, and Han.

I really like David P's idea that represented the fourier transform as
an infinite product. I am still working my way through the proofs by
David U.

Let me tell you about my attempts to grapple with this problem. I had
an idea for a proof that has not been written out to show that p cannot
be greater than 1. (David U's proof is shorter.) Maybe later I can
post this proof. I also looked at the function

f(x) = Exp( -1/(x-1)/(1-x) ) when abs(x) <=1 and f(x) = 0 otherwise.

This function is infinitely differentiable and compactly supported. I
numerically caculated the log(abs(g(s))) for s = 1, ... 40 and got

-3.15, -8.143, -5.982, -6.567, -7.465, -8.697, -12.108, -10.05, \
-9.968, -10.223, -10.637, -11.173, -11.848, -12.761, -14.579, \
-14.156, -13.671, -13.634, -13.766, -13.997, -14.302, -14.675, \
-15.121, -15.666, -16.377, -17.533, -18.628, -17.468, -17.204, -17.162,
\
-17.229, -17.367, -17.557, -17.793, -18.072, -18.396, -18.773, -19.222,
\
-19.784, -20.584

These values were approximately equal to -3.22*sqrt(s), so g(s) seems
to be order exp(-abs(1/s^p)) for any p < 1/2.

Of course, these are just calculations and they are not a proof.

.

Relevant Pages

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