Appoximate a non-analytical function?

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Approximating a non-analytic function near a point raises questions about the nature of the function, particularly its continuity and differentiability. The discussion highlights the complexity of determining whether solutions to ordinary differential equations (ODEs) or partial differential equations (PDEs) are analytic, linking it to Hilbert's 19th problem regarding the regularity of solutions in calculus of variations. It is noted that Taylor series may fail for certain infinitely differentiable functions, prompting inquiries into alternative approximation methods. The conversation suggests that the context of the approximation—whether numerical or theoretical—plays a crucial role in finding a solution. Overall, the challenge of approximating non-analytic functions remains an open and complex topic in mathematical analysis.
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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!
 
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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.
 
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.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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