Positive-definite function

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

1 In dynamical systems
2 In complex analysis
3 See also
4 References

[edit] In dynamical systems

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 $x\in D$ .^[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.

[edit] In complex analysis

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

f:R → C

such that for any real numbers

x₁, ..., x_n

the n×n matrix A with entries

a_ij = f(x_i − x_j)

is 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^{[citation needed]}.

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).

[edit] See also

Positive-definite matrix

[edit] References

^ Verhulst, Ferdinand (1996). Nonlinear Differential Equations and Dynamical Systems, 2nd ed., Springer. ISBN 3-540-60934-2.
^ Hahn, Wolfgang (1967). Stability of Motion. Springer.