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Let $f$ and $g$ be distributions on $\mathbb{R}$ with compact support. Do we have

$\inf (\textrm{supp}(f*g)) = \inf (\textrm{supp}(f)) + \inf (\textrm{supp}(g))$

Where 'supp' denotes de the support of a distribution ?

The left term is obviously greater than the right one. But the other inequality seems trickier to me (I guess it's always easier to prove that things are equal to zero). Any help in finding either a counterexample, proof, or reference for that statement would be appreciated. Thanks.

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    @Rasmus : That is indeed what I meant.2011-05-06

1 Answers 1

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You might wish to take a look at the Titchmarsh Convolution Theorem and its generalization due to Lions, which states that your equation holds. A proof can be found in Hörmander's Analysis of linear partial differential operators I, section 4.3.

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    Thanks, this is exactly what I was looking for.2011-05-06