Re: positive definite functions

David C. Ullrich wrote:
> On 8 Aug 2005 06:41:50 GMT, israel@xxxxxxxxxxx (Robert Israel) wrote:
>
> >In article <john.lord-3786F5.16274706082005@xxxxxxxxxxxxxx>,
> >john f. lord <john.lord@xxxxxxxxxxxx> wrote:
> >>In article <soj6f119lvfg6h7mqg2q4fdmnsei2k1aq8@xxxxxxx>,
> >> David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:
> >
> >>> Anyway, the interesting thing here is Bochner's theorem: A function
> >>> on R is the Fourier transform of a Borel probability measure _if and
> >>> only if_ it's continuous, positive definite and equals 1 at the
> >>> origin. (Similarly in more general settings, for example locally
> >>> compact abelian groups at least.)
> >
> >>Now what my question was meant to be: are there ways to verify (at least
> >>for some classes of functions) if a function is positive definite,
> >>without calculation of its Fourier transform or of the determinants in
> >>the definition of positive-definiteness, as both in general are
> >>prohibitively complicated?
> >
> >In some cases, yes. For example, any "autocorrelation" function of the form
> >f(x) = int_{-infty}^infty g(t+x) conjugate(g(t)) dt (where g is square-integrable)
> >is positive definite.
>
> Given f, how do you tell whether there exists such a g?

In general, you don't (unless by calculating the Fourier transform).
What
I meant was that f might be initially given in this form, or as a sum
of functions of this form.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

.

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