Appoximate a non-analytical function?

[zeebek]

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!

 

[JeSuisConf]

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.

 

[zeebek]

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.