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Pointwise Convergence For Induced Distributions

  1. Feb 5, 2012 #1
    1. The problem statement, all variables and given/known data

    If [itex] U \subseteq \mathbb R^n [/itex] find a sequence of locally integrable functions [itex] f_n \in L^1_{\text{loc}}(U) [/itex] which converge pointwise, but whose induced distributions
    [tex] \langle f_n, \cdot \rangle: C_c^\infty(U) \to \mathbb R, \qquad \langle f_n, \phi \rangle = \int_U f_n \phi [/tex]
    do not converge in the weak* topology.

    3. The attempt at a solution
    It would seem like the typical examples of "integral breaking" functions do not work here. In particular, I have tried examples of functions which spread ([itex] \frac1n\chi_{(0,n)} [/itex]), congregate ([itex] \frac n2 \chi_{[-1/n,1/n]} [/itex]) et cetera, but these do not seem to work. In particular, I think that it is because these choice of functions are not only locally integrable, but are integrable on all of [itex] \mathbb R [/itex] and converge weakly in [itex] L^p [/itex]; in fact, each of the above converge to the Heaviside and delta distributions respectively. Any ideas would be useful.
     
  2. jcsd
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