Re: Fractional Transforms

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.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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