Re: Fractional Transforms
- From: David C. Ullrich <dullrich@xxxxxxxxxxx>
- Date: Mon, 15 Dec 2008 14:53:46 -0600
On Mon, 15 Dec 2008 14:05:57 -0600, Robert Israel
<israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
David C. Ullrich <dullrich@xxxxxxxxxxx> writes:
A few days ago I decided there was no point, considering the audience,
in pointing out that the existence of a T with L = T^2 is clearly
impossible if we interpret things very strictly. We have
L: X -> Y where X and Y are very different spaces of functions.
If T were a functional square root of L then we'd need to have
somehow T:X -> Z and also T:Z -> Y; if we take T to have
a well-defined domain and co-domain then it follows that
X = Y. Of course this doesn't quite rule out the existence
of some kernel that does the job at least formally.
On the other hand there are spaces X such that L: X -> X.
For example, let X be the space of continuous functions f on (0,infty)
such that x^(1/2) f(x) is bounded.
Heh, very good - I considered the possibility of such an X existing
but didn't think of this simple example.
(Readers who are wondering why this X has that property
should contemplate the Laplace transform of 1/t^(1/2)...)
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.
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