Re: positive definite functions

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. And a linear combination of positive definite functions with
nonnegative coefficients is positive definite. The best way to prove that the
matrices are positive semidefinite is not to use determinants but to use the
following characterization: the Hermitian matrix A is positive semidefinite
if x^* A x >= 0 for all vectors x (where ^* denotes the conjugate transpose).

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

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Relevant Pages

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  • Re: positive definite functions
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