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In mathematics, the term positive-definite function may refer to a couple of different concepts.
## In dynamical systems

## In complex analysis and statistics

## See also

## References

A real-valued, continuously differentiable function f is positive definite on a neighborhood of the origin, D, if $f(0)=0$ and $f(x)>0$ for every non-zero $xin\; D$.

A function is negative definite if the inequality is reversed. A function is semidefinite if the strong inequality is replaced with a weak ($geq$ or $leq$) one.

A positive-definite function of a real variable x is a complex-valued function

- f:R → C

such that for any real numbers

- x
_{1}, ..., x_{n}

the n×n matrix A with entries

- a
_{ij}= f(x_{i}− x_{j})

is a positive semi-definite matrix. It is usual to restrict to the case in which f(−x) is the complex conjugate of f(x), making the matrix A Hermitian.

If a function f is positive semidefinite, we find by taking n = 1 that

- f(0) ≥ 0.

By taking n=2 and recognising that a positive-definite matrix has a positive determinant we get

- f(x − y)f(y − x) ≤ f(0)
^{2}

which implies

- |f(x)| ≤ f(0).

Positive-definiteness arises naturally in the theory of the Fourier transform; it is easy to see directly that to be positive-definite is a necessary condition on f, for it to be the Fourier transform of a function g on the real line with g(y) ≥ 0.

The converse result is Bochner's theorem, stating that a continuous positive-definite function on the real line is the Fourier transform of a (positive) measure.

This result generalizes to the context of Pontryagin duality, with positive-definite functions defined on any locally compact abelian topological group. Positive-definite functions also occur naturally in the representation theory of groups on Hilbert spaces (i.e. the theory of unitary representations).

In statistics, the theorem is usually applied to real functions; the matrix A is then multiplied by a scalar to give a covariance matrix, which must be positive definite. In a statistical context, one does not usually use Fourier terminology and instead one states that f(x) is the characteristic function of a symmetric PDF.

- Z. Sasvári, Positive Definite and Definitizable Functions, Akademie Verlag, 1994

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Last updated on Friday September 05, 2008 at 14:58:12 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday September 05, 2008 at 14:58:12 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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