Positive-definite function
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In mathematics, a positive-definite function of a real variable x is a function
- f:R → C
such that for any real numbers
- x1, ...,xn
the n×n matrix A with entries
- aij = f(xi − xj)
is positive semi-definite. It is usual to restrict to the case in which f(−x) is the complex conjugate of f(x), making the matrix A Hermitian.
For example, taking n = 1 we must have
- f(0) ≥ 0
and taking n = 2 the product
- f(x − y)f(y − x) ≤ f(0)2;
therefore necessarily
- |f(x)| ≤ f(0).
This condition 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 generalises 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).