Uniformly bounded derivatives implies globally analytic: Difference between revisions

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:

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

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.

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 ==Statement==
+===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:
@@ Line 6: / Line 8: @@
 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>.
+==Examples==
+The functions <math>\exp, \sin, \cos</math> all fit this description.