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FourierTransform and removable singularities


It seems to me that Mathematica 5.2 is not careful enough when doing
Fourier transforms of functions with delta functions at removable
singularities: if you call

    FourierTransform[DiracDelta[t], t, w]

you get the right answer,

    1/Sqrt[2*Pi]

But if you call something of the sort of

    FourierTransform[DiracDelta[t]*(Sin[t]/t), t, w]

which has a removable singularity at the point where the Dirac delta
function acts, the answer is zero, which is wrong.

Does anyone know how to resolve this by reformulating the problem? (a
workaround)

Cheers!
Roman.



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