In mathematics, a positive-definite function is, depending on the context, either of two types of function.
Let
R
C
A function
f:R\toC
A=\left(aij
n~, | |
\right) | |
i,j=1 |
aij=f(xi-xj)
is a positive semi-definite matrix.
By definition, a positive semi-definite matrix, such as
A
In particular, it is necessary (but not sufficient) that
f(0)\geq0~, |f(x)|\leqf(0)
(these inequalities follow from the condition for n = 1, 2.)
A function is negative semi-definite if the inequality is reversed. A function is definite if the weak inequality is replaced with a strong (<, > 0).
If
(X,\langle ⋅ , ⋅ \rangle)
gy\colonX\toC
x\mapsto\exp(i\langley,x\rangle)
y\inX
u\inCn
x1,\ldots,xn
u*
(gy) | |
A |
u =
n | |
\sum | |
j,k=1 |
\overline{uk}uj
i\langley,xk-xj\rangle | |
e |
=
n | |
\sum | |
k=1 |
\overline{uk}
i\langley,xk\rangle | |
e |
n | |
\sum | |
j=1 |
uj
-i\langley,xj\rangle | |
e |
=\left|
n | |
\sum | |
j=1 |
\overline{uj}
i\langley,xj\rangle | |
e |
\right|2 \ge0.
\cos(x)=
1 | |
2 |
(ei+e-)=
1 | |
2 |
(g1+g-1).
One can create a positive definite function
f\colonX\toC
f\colon\R\toC
X
\phi\colonX\to\R
f*:=f\circ\phi
u*
(f*) | |
A |
u =
n | |
\sum | |
j,k=1 |
\overline{uk}uj
*(x | |
f | |
k |
-xj)=
n | |
\sum | |
j,k=1 |
\overline{uk}ujf(\phi(xk)-\phi(xj))=u*\tilde{A}(f)u \ge0,
\tilde{A}(f)=(f(\phi(xi)-\phi(xj))=f(\tilde{x}i-\tilde{x}j))i,
\tilde{x}k:=\phi(xk)
\phi
See main article: Bochner's theorem.
Positive-definiteness arises naturally in the theory of the Fourier transform; it can be seen directly that to be positive-definite 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 positive-definite function on the real line is the Fourier transform of a (positive) measure.[2]
In statistics, and especially Bayesian statistics, the theorem is usually applied to real functions. Typically, n scalar measurements of some scalar value at points in
Rd
In this context, Fourier terminology is not normally used and instead it is stated that f(x) is the characteristic function of a symmetric probability density function (PDF).
See main article: Positive-definite function on a group.
One can define positive-definite functions on any locally compact abelian topological group; Bochner's theorem extends to this context. Positive-definite functions on groups occur naturally in the representation theory of groups on Hilbert spaces (i.e. the theory of unitary representations).
Alternatively, a function
f:\realsn\to\reals
f(0)=0
f(x)>0
x\inD
Note that this definition conflicts with definition 1, given above.
In physics, the requirement that
f(0)=0
. Salomon Bochner . Lectures on Fourier integrals . registration . Princeton University Press . 1959.