Re: Fourier Transform, Smooth Functions

On 27 Feb 2006 13:13:14 -0800, "irchans" <infinitgames@xxxxxxxxx>
wrote:

Hi David,

I think I understand the both the argument that p=1 is impossible
and the exposition of Petry's example, but I still want to write out
all the details for myself.

Ok.

I just started reading Rudin's section on
quasi-analytic functions (Denjoy-Carleman theorem) and I'm sure it will
help me understand the problem more deeply. Thank you very much for
the reference!

It's a worthwhile theorem, but I don't think that it settles the
question you asked. At least I don't see how it does:

Assume that g(s) is of order exp(-|s|^p), where p < 1. That implies
bounds on the absolute value of the derivatives f^(n). The D-C
theorem says that the class of all f satisfying those bounds is
not quasi-analytic, hence contains a function F with compact
support. But I don't see how it follows that the Fourier transform
of F is of order exp(-|s|^p).

Cheers,
Irchans


************************

David C. Ullrich
.