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Distributions with compact support are tempered

  1. Oct 18, 2011 #1
    1. The problem statement, all variables and given/known data
    Let [itex]F \in \mathcal E'(\mathbf R)[/itex]. Prove that [itex]F \in \mathcal S'(\mathbf R)[/itex].

    2. The attempt at a solution
    Since [itex]F \in \mathcal E'(\mathbf R)[/itex], there exists a continuous function [itex]f \colon \mathbf R \to \mathbf R[/itex] and a nonnegative integer [itex]k[/itex] such that for every [itex]\varphi \in \mathcal E(\mathbf R)[/itex],

    [tex]\langle F, \varphi \rangle = \int_{-\infty}^\infty f(x) \varphi^{(k)}(x) \, dx[/tex]

    To prove that [itex]F \in \mathcal S'(\mathbf R)[/itex], I need to prove that if [itex]\varphi_n \to 0[/itex] in [itex]\mathcal S(\mathbf R)[/itex], then [itex]\langle F, \varphi_n \rangle \to 0[/itex]. Using the above integral representation for [itex]F[/itex], I was hoping to pass the limit under the integral sign using the dominated convergence theorem but I cannot think of any dominating integrable function. I don't know what else to do here. Any tips?
  2. jcsd
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