🚀 go-pugleaf

In article <27CD919E.5D23@deneva.sdd.trw.com> jumper@spf.trw.com (Greg Jumper) writes:
[The attribution for the original posting seems to have been lost in the
follow-up-- DH]
>>  As a side thought, if you know all the derivatives of a function at a
>>  single point in time, you can find the value of the function for all times.
>
>This statement is only true if the function is so-called "real-analytic,"
>which is more restrictive than even "smooth" (C-infinity).  Admittedly,
>functions which are not real-analytic are "pathological," particularly in a
>"real-world" setting.
>
>Ironically (since the subject is sampling theory), it turns out that the
>examples of functions whose Taylor series representations do not converge,
>even though all their derivatives exist everywhere, are constructed using
>Fourier methods -- essentially by taking advantage of "Gibb's phenomenon" to
>produce non-convergence.  (Even "almost everywhere", if I remember correctly.)
>
>An interesting counter-example!

As is         
              / exp(-1/x^2)    x <> 0
      f(x) = <
              \ 0              x = 0


The original statement, even as qualified, is not precisely true, as it
assumes an infinite radius of convergence for the Taylor series.

*D*