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From: jumper@spf.trw.com (Greg Jumper)
Newsgroups: comp.dsp,sci.math
Subject: Re: resampling problem
Message-ID: <27CD919E.5D23@deneva.sdd.trw.com>
Date: 28 Feb 91 23:26:21 GMT
References: <1991Feb13.234510.22488@nuchat.sccsi.com> <25328@netcom.COM> <gah.667695466@valine>
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Organization: TRW Data Systems Center, Redondo Beach, CA
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>  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!



                                       Greg Jumper
                                       TRW Data Systems Center

                                       jumper@spf.trw.com