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Re: heat extension commuting with multiplication
Posted:
Jun 21, 2011 6:30 AM
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Prabhu <pjanakir1978@gmail.com> writes:
> Let f and g be two complex-valued L^2 functions on the plane. Let F, G
> and H be the heat extensions of f, g and fg respectively.
>
> Are there ever non-trivial f and g such that H = FG ?
>
> (I believe the answer is NO; but how does one establish this?)
>
> Prabhu
What about an argument by Taylor expansions? If you assume that f,g are
analytic (which you should be able to do by starting from some t=eps>0
otherwise) the series expansions of F, G are determined by the series of
f,g. Now H must be equal to both the heat extension of the series of fg
and the product series FG. This argument is quite technical, but I
think you should be able to derive that those series must be trivial.
Nicolas
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