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Harnack's Principle question

  1. Jan 26, 2009 #1
    Ahlfors version of this theorem says that a sequence of harmonic functions {Un} tends UNIFORMLY to infinity on compact subsets, or tends to a harmonic limit function uniformly on compact sets.

    Can someone please clarify what tending uniformly to infinity means?

    In particular, it seems like a set of harmonic {Un} where Uk = k (such that each function is constant) tends non-uniformly to infinity.

    So I must be missing something somewhere.

    thanks
     
  2. jcsd
  3. Jan 30, 2009 #2
    [itex]f_n:K\to\mathbb{C}[/itex] ([itex]n=1,2,3\ldots[/itex]) converges towards infinity uniformly, if for all [itex]R>0[/itex] there exists [itex]N\in\mathbb{N}[/itex] such that [itex]|f_n(z)|>R[/itex] for all [itex]z\in K[/itex] and [itex]n\geq N[/itex].

    Simply extend [itex][0,\infty[[/itex] to [itex][0,\infty][/itex] (with topology homeomorphic with [itex][0,1][/itex]), and threat [itex]\infty[/itex] as a constant so that a sequence of functions can converge uniformly towards the corresponding constant function.

    In point-wise convergence to infinity would mean that for each [itex]z\in K[/itex] and [itex]R>0[/itex] there exists [itex]N\in\mathbb{N}[/itex] such that [itex]|f_n(z)|>R[/itex] for all [itex]n\geq N[/itex].

    Constant functions surely converge uniformly if they converge somehow.
     
  4. Jan 30, 2009 #3
    Thanks. That definition clears things up greatly.
     
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