Appoximate a non-analytical function?

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In summary, the conversation discusses the possibility of approximating a non-analytic function near a point and whether there is a criterion to determine if a solution to a differential equation is analytic. The conversation also references the Picard-Lindelöf theorem and Hilbert's 19th problem. The question of approximating non-analytic functions near a point is raised, with a reference to a type of function that is infinitely differentiable but does not have a Taylor series at certain points. The possibility of finding a method to approximate such functions near a point is also discussed.
  • #1
zeebek
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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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  • #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.
 
  • #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.
 

1. How can I approximate a non-analytical function?

There are several methods for approximating a non-analytical function, such as using numerical integration or regression techniques. These methods involve dividing the function into smaller, simpler parts and using mathematical algorithms to estimate the behavior of the function.

2. What is the difference between an analytical and a non-analytical function?

An analytical function, also known as a mathematically defined function, can be expressed using a finite number of algebraic operations and mathematical functions. On the other hand, a non-analytical function cannot be expressed in this way and may require numerical methods for approximation.

3. Can I use an analytical method to approximate a non-analytical function?

No, analytical methods are only applicable to analytical functions. Attempting to use an analytical method on a non-analytical function will not yield accurate results.

4. What are some common applications for approximating non-analytical functions?

Approximating non-analytical functions is commonly used in fields such as engineering, physics, and economics. It can be used to estimate physical phenomena, model complex systems, and make predictions based on data.

5. Are there any limitations to approximating non-analytical functions?

Yes, there can be limitations to the accuracy and reliability of approximating non-analytical functions. The quality of the approximation depends on the chosen method and the complexity of the function. Additionally, approximations may become less accurate as the function approaches singularities or extreme values.

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