Difference between revisions of "Uniformly bounded derivatives implies globally analytic"

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==Statement==
 
==Statement==
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===Global statement===
  
 
Suppose <math>f</math> is an infinitely differentiable function on <math>\R</math> such that, for any fixed <math>a,b \in \R</math>, there is a constant <math>C</math> (possibly dependent on <math>a,b</math>) such that for all nonnegative integers <math>n</math>, we have:
 
Suppose <math>f</math> is an infinitely differentiable function on <math>\R</math> such that, for any fixed <math>a,b \in \R</math>, there is a constant <math>C</math> (possibly dependent on <math>a,b</math>) such that for all nonnegative integers <math>n</math>, we have:
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Then, <math>f</math> is a [[globally analytic function]]: the [[Taylor series]] of <math>f</math> about any point in <math>\R</math> converges to <math>f</math>. In particular, the Taylor series of <math>f</math> about 0 converges to <math>f</math>.
 
Then, <math>f</math> is a [[globally analytic function]]: the [[Taylor series]] of <math>f</math> about any point in <math>\R</math> converges to <math>f</math>. In particular, the Taylor series of <math>f</math> about 0 converges to <math>f</math>.
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==Examples==
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The functions <math>\exp, \sin, \cos</math> all fit this description.

Revision as of 14:51, 7 July 2012

Statement

Global statement

Suppose f is an infinitely differentiable function on \R such that, for any fixed a,b \in \R, there is a constant C (possibly dependent on a,b) such that for all nonnegative integers n, we have:

|f^{(n)}(x)| \le C \ \forall x \in [a,b]

Then, f is a globally analytic function: the Taylor series of f about any point in \R converges to f. In particular, the Taylor series of f about 0 converges to f.

Examples

The functions \exp, \sin, \cos all fit this description.