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

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Revision as of 14:44, 8 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.