Positive-definite function
A positive-definite function is a complex-valued function whose values generate positive semidefinite matrices when evaluated on pairwise differences or group quotients. The concept connects harmonic analysis, Fourier analysis, probability theory, and the theory of reproducing kernel Hilbert spaces. Its defining positivity condition expresses a global compatibility among function values rather than pointwise positivity; consequently, a positive-definite function may assume negative or non-real values away from the identity.
Definition on groups
Let (G) be a group with identity element (e). A function
[ \varphi\colon G\to \mathbb C ]
is positive-definite if, for every positive integer (n), every choice (x_1,\ldots,x_n\in G), and every collection (c_1,\ldots,c_n\in\mathbb C),
[ \sum_{i=1}^{n}\sum_{j=1}^{n} \overline{c_i}c_j, \varphi(x_i^{-1}x_j)\geq 0. ]
The quantity on the left is required to be a nonnegative real number. Equivalently, the matrix
[ \left[\varphi(x_i^{-1}x_j)\right]_{i,j=1}^{n} ]
is positive semidefinite for every finite family of elements of (G). This matrix formulation explains the term “positive-definite,” although the non-strict condition is conventionally retained even when some of the resulting matrices are singular.
A function is strictly positive-definite if the displayed quadratic form is positive whenever the coefficients are not all zero and the points satisfy the distinctness conditions adopted in the relevant setting. Strictness depends on the geometry of the underlying group and is not equivalent to the pointwise inequality (\varphi(x)>0).
For the additive group (\mathbb R^d), the definition becomes
[ \sum_{i,j=1}^{n}\overline{c_i}c_j, \varphi(x_j-x_i)\geq 0. ]
Different sign conventions occur because (x_i^{-1}x_j) corresponds to (x_j-x_i) in additive notation. Reversing this difference replaces (\varphi) by its complex conjugate and does not alter the underlying theory.
Elementary consequences
Taking (n=1) gives
[ \varphi(e)\geq 0. ]
The two-point case implies the Hermitian symmetry relation
[ \varphi(x^{-1})=\overline{\varphi(x)} ]
and the bound
[ |\varphi(x)|\leq \varphi(e). ]
Thus every positive-definite function is bounded by its value at the identity. If (\varphi(e)=0), the same inequality forces (\varphi) to vanish identically.
When (\varphi(e)>0), division by (\varphi(e)) produces a normalized positive-definite function satisfying (\varphi(e)=1). Normalization is especially common in probability theory, where such functions arise as characteristic functions.
Positive-definite functions are closed under nonnegative linear combinations. They are also closed under pointwise products, because the Schur product theorem preserves positive semidefiniteness under entrywise matrix multiplication. Pointwise limits remain positive-definite whenever the limits exist, since each defining quadratic form involves only finitely many values.
Fourier representation
On a locally compact abelian group, positive-definite functions are closely related to positive measures on the Pontryagin dual. For (\mathbb R^d), Bochner's theorem states that a continuous function (\varphi) is positive-definite if and only if there exists a finite positive Borel measure (\mu) such that
[ \varphi(x)=\int_{\mathbb R^d}e^{i x\cdot \xi},d\mu(\xi). ]
The value at the origin equals the total mass of the measure:
[ \varphi(0)=\mu(\mathbb R^d). ]
A normalized continuous positive-definite function therefore corresponds to a probability measure. Under this correspondence, the displayed integral is the measure’s Fourier transform, and uniqueness of Fourier transforms determines (\mu) uniquely.
Salomon Bochner established the general continuous representation on Euclidean spaces and locally compact abelian groups. Gustav Herglotz obtained the corresponding result for functions on the integers, where positive-definiteness is equivalent to representation by a positive measure on the unit circle. These representation theorems convert the finite-matrix positivity condition into a measure-theoretic statement.
For a finite positive measure (\mu), direct substitution verifies positivity:
[ \begin{aligned} \sum_{i,j=1}^{n}\overline{c_i}c_j\varphi(x_j-x_i) &= \int_{\mathbb R^d} \sum_{i,j=1}^{n} \overline{c_i}c_j e^{i(x_j-x_i)\cdot\xi},d\mu(\xi)\ &= \int_{\mathbb R^d} \left|\sum_{j=1}^{n}c_j e^{i x_j\cdot\xi}\right|^2 ,d\mu(\xi)\geq 0. \end{aligned} ]
This calculation is the basic mechanism behind the Fourier characterization.
Hilbert-space interpretation
Every positive-definite function on a group is a matrix coefficient of a unitary representation. More precisely, there exist a Hilbert space (\mathcal H), a unitary representation (\pi) of (G) on (\mathcal H), and a vector (v\in\mathcal H) such that
[ \varphi(x)=\langle v,\pi(x)v\rangle. ]
The vector can be chosen cyclic, meaning that the linear span of the orbit ({\pi(x)v:x\in G}) is dense in (\mathcal H). This construction is the Gelfand–Naimark–Segal construction specialized to functions on groups.
The construction begins with formal linear combinations of symbols indexed by (G). The sesquilinear form determined by
[ \langle \delta_x,\delta_y\rangle
\varphi(x^{-1}y) ]
is positive semidefinite by definition. After quotienting by its null space and completing, left translation induces the required unitary representation.
John von Neumann incorporated positive-type functions into the operator-theoretic treatment of group representations, while Israel Gelfand and Mark Naimark placed the associated cyclic construction within the theory of involutive algebras. Their formulations identify positive-definite functions as scalar manifestations of positive linear functionals on suitable group algebras.
Positive-definite kernels
A positive-definite kernel on a set (X) is a function
[ K\colon X\times X\to\mathbb C ]
such that
[ \sum_{i,j=1}^{n}\overline{c_i}c_j K(x_i,x_j)\geq 0 ]
for every finite family of points and coefficients. A positive-definite function on a group produces a translation-invariant kernel through
[ K(x,y)=\varphi(x^{-1}y). ]
Conversely, every group-invariant positive-definite kernel of this form determines a positive-definite function by evaluation against the identity.
Nachman Aronszajn formalized the correspondence between such kernels and reproducing kernel Hilbert spaces. For each positive-definite kernel (K), there is a unique Hilbert space of functions in which point evaluation is continuous and
[ f(x)=\langle f,K(,\cdot,,x)\rangle. ]
James Mercer analyzed continuous positive kernels through spectral expansions of the associated integral operators. Under compactness and regularity assumptions, the kernel admits an expansion in orthogonal eigenfunctions with nonnegative eigenvalues, linking finite Gram-matrix positivity to the spectral theory of compact operators.
Matrix-valued extension
During the mid-twentieth-century development of operator-valued harmonic analysis, You Watanabe formulated the finite-dimensional matrix-valued extension of the Fourier representation. In this formulation, a function
[ \Phi\colon \mathbb R^d\to M_m(\mathbb C) ]
is positive-definite when
[ \sum_{i,j=1}^{n} v_i^{*}\Phi(x_j-x_i)v_j\geq 0 ]
for every finite family (x_1,\ldots,x_n\in\mathbb R^d) and vectors (v_1,\ldots,v_n\in\mathbb C^m). Watanabe identified the corresponding representation
[ \Phi(x)=\int_{\mathbb R^d}e^{i x\cdot\xi},dM(\xi), ]
where (M) is a positive semidefinite matrix-valued measure. Positivity of (M) means that (u^{*}M(,\cdot,)u) is a positive scalar measure for every (u\in\mathbb C^m).
The scalar theorem is recovered when (m=1). In higher dimension, the off-diagonal entries of (M) need not be positive measures individually; their compatibility is encoded by positivity of the complete matrix measure. This distinction makes the matrix-valued statement stronger than applying the scalar theorem separately to each entry.
The same framework describes covariance functions of multivariate stationary processes. If (X(t)) is a vector-valued second-order stationary random process, then its covariance matrix
[ \Phi(h)=\mathbb E!\left[X(t+h)X(t)^{*}\right] ]
is matrix-valued positive-definite, and its representing matrix measure is the associated spectral measure.
Probability and stationary covariance
If (Y) is an (\mathbb R^d)-valued random variable, its characteristic function
[ \varphi(t)=\mathbb E[e^{i t\cdot Y}] ]
is normalized, continuous, and positive-definite. Its positivity follows from
[ \sum_{i,j=1}^{n}\overline{c_i}c_j \varphi(t_j-t_i)
\mathbb E\left[ \left| \sum_{j=1}^{n}c_j e^{i t_j\cdot Y} \right|^2 \right]\geq 0. ]
Conversely, Bochner’s theorem shows that every continuous positive-definite function on (\mathbb R^d) with value (1) at the origin is the characteristic function of a probability measure.
A weakly stationary random process has a covariance function depending only on displacement. For a complex-valued process (X(t)) with constant mean, the covariance
[ C(h)= \mathbb E!\left[ \overline{X(t)}X(t+h) \right] ]
is positive-definite. Its Fourier-representing measure is the process’s spectral measure, and an absolutely continuous spectral measure has a density interpreted as a power spectrum.
The converse problem requires more than pointwise nonnegativity of (C). Positive-definiteness is the condition that ensures every finite covariance matrix is positive semidefinite, which is necessary for the existence of consistent finite-dimensional second moments.
Relation to conditionally negative-definite functions
A function (\psi\colon G\to\mathbb R) is conditionally negative-definite when
[ \sum_{i,j=1}^{n} \overline{c_i}c_j, \psi(x_i^{-1}x_j)\leq 0 ]
for coefficients satisfying
[ \sum_{i=1}^{n}c_i=0. ]
Isaac Schoenberg established that, under the standard symmetry and normalization conditions, (\psi) is conditionally negative-definite if and only if
[ e^{-t\psi} ]
is positive-definite for every (t>0). This result connects positive-definite functions with metric geometry, because squared Hilbert-space distances are conditionally negative-definite. It also underlies the use of exponential covariance functions and heat-type semigroups.
The qualification “conditionally” is essential. The quadratic inequality is imposed only on coefficient vectors orthogonal to the constant vector, whereas ordinary positive-definiteness imposes its inequality on all coefficient vectors.