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

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(Created page with "==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...")
 
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<math>|f^{(n)}(x)| \le C \ \forall x \in [a,b]</math>
 
<math>|f^{(n)}(x)| \le C \ \forall x \in [a,b]</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 0.
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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>.

Revision as of 14:51, 7 July 2012

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.