Convolution of distributions - Part 2

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Convolution of distributions - Part 2
Prof. S Kesavan
Department of Mathematics
The Institute of Mathematical Sciences
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I don't understand the step at 17:16, the one where you get (T* phi tilde)* (phi * psi). How can you perform associativity if psi doesn't necessarily have compact support, and if such associativity is exactly what we're trying to prove?

thephysicistcuber