> 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.