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

Revision as of 14:44, 8 July 2012

Statement

Global statement

Suppose is an infinitely differentiable function on such that, for any fixed , there is a constant (possibly dependent on ) such that for all nonnegative integers , we have:

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

Examples

The functions all fit this description.