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Appoximate a non-analytical function?

  1. Jun 2, 2010 #1
    Appoximate a non-analytic function?

    Dear all,

    is there a way to do that near a point?

    Also, for a given ODE or even PDE, is there a criterion to show whether its solution is analytic? Is it a proper question in fact?

    Thanks!
     
    Last edited: Jun 3, 2010
  2. jcsd
  3. Jun 3, 2010 #2
    Not sure what you mean about approximating a non-analytical function near a point. It depends what you mean, and what the function is. Is it continuous? And are you only concerned about numerics here - or what is the context?

    Regarding the DE/PDE's here's a starting point for you:

    http://en.wikipedia.org/wiki/Picard–Lindelöf_theorem

    If you ever find a way to do it in general for PDE's, you're a genius.
     
  4. Jun 3, 2010 #3
    Thanks you for your reply. Basically my second question turned out to be partly a rephrase of Hillberts 19th problem as I just found out:

    "Are the solutions of regular problems in the calculus of variations always analytic?"
    Here is an overview http://math.univ-lyon1.fr/~clarke/Clarke_Regularity.pdf

    The first question: for a certain type of functions Taylor series do not work at some points even though they are infitely differentiable there like this one
    http://planetmath.org/?op=getobj&from=objects&id=3081
    So the question is is there a method to approximate such a function near a point still.
     
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