In mathematics, the term positivedefinite function may refer to a couple of different concepts.
Contents

In dynamical systems 1

In analysis 2

Bochner's theorem 2.1

Generalisation 2.2

References 3

Notes 4

External links 5
In dynamical systems
A realvalued, continuously differentiable function f is positive definite on a neighborhood of the origin, D, if f(0)=0 and f(x)>0 for every nonzero x\in D.^{[1]}^{[2]}
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.
In analysis
A positivedefinite function of a real variable x is a complexvalued function f:R → C such that for any real numbers x_{1}, ..., x_{n} the n×n matrix

A = (a_{i,j})_{i,j=1}^n~, \quad a_{i,j} = f(x_i  x_j)
is positive semidefinite (which requires A to be Hermitian; therefore f(x) is the complex conjugate of f(x)).
In particular, it is necessary (but not sufficient) that

f(0) \geq 0~, \quad f(x) \leq f(0)
(these inequalities follow from the condition for n=1,2.)
Bochner's theorem
Positivedefiniteness arises naturally in the theory of the Fourier transform; it is easy to see directly that to be positivedefinite it is sufficient for f 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 any continuous positivedefinite function on the real line is the Fourier transform of a (positive) measure.^{[3]}
Applications
In statistics, and especially Bayesian statistics, the theorem is usually applied to real functions. Typically, one takes n scalar measurements of some scalar value at points in R^d and one requires that points that are closely separated have measurements that are highly correlated. In practice, one must be careful to ensure that the resulting covariance matrix (an nbyn matrix) is always positive definite. One strategy is to define a correlation matrix A which is then multiplied by a scalar to give a covariance matrix: this must be positive definite. Bochner's theorem states that if the correlation between two points is dependent only upon the distance between them (via function f()), then function f() must be positive definite to ensure the covariance matrix A is positive definite. See Kriging.
In this context, one does not usually use Fourier terminology and instead one states that f(x) is the characteristic function of a symmetric PDF.
Generalisation
One can define positivedefinite functions on any locally compact abelian topological group; Bochner's theorem extends to this context. Positivedefinite functions on groups occur naturally in the representation theory of groups on Hilbert spaces (i.e. the theory of unitary representations).
References

Christian Berg, Christensen, Paul Ressel. Harmonic Analysis on Semigroups, GTM, Springer Verlag.

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

Wells, J. H.; Williams, L. R. Embeddings and extensions in analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. SpringerVerlag, New YorkHeidelberg, 1975. vii+108 pp.
Notes

^ Verhulst, Ferdinand (1996). Nonlinear Differential Equations and Dynamical Systems (2nd ed.). Springer.

^ Hahn, Wolfgang (1967). Stability of Motion. Springer.

^
External links

Hazewinkel, Michiel, ed. (2001), "Positivedefinite function",
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