In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations. Since then, positive-definite functions and their various analogues and generalizations have arisen in diverse parts of mathematics. They occur naturally in Fourier analysis, probability theory, operator theory, complex function-theory, moment problems, integral equations, boundary-value problems for partial differential equations, machine learning, the embedding problem, information theory, and other areas.
Definition Let X {\displaystyle {\mathcal {X}}} be a nonempty set, sometimes referred to as the index set. A symmetric function K : X × X → R {\displaystyle K:{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} } is called a positive-definite (p.d.) kernel on X {\displaystyle {\mathcal {X}}} if
holds for all x 1 , … , x n ∈ X {\displaystyle x_{1},\dots ,x_{n}\in {\mathcal {X}}} , n ∈ N , c 1 , … , c n ∈ R {\displaystyle n\in \mathbb {N} ,c_{1},\dots ,c_{n}\in \mathbb {R} } . In probability theory, a distinction is sometimes made between positive-definite kernels, for which equality in (1.1) implies c i = 0 ( ∀ i ) {\displaystyle c_{i}=0\;(\forall i)} , and positive semi-definite (p.s.d.) kernels, which do not impose this condition. Note that this is equivalent to requiring that every finite matrix constructed by pairwise evaluation, K i j = K ( x i , x j ) {\displaystyle \mathbf {K} _{ij}=K(x_{i},x_{j})} , has either entirely positive (p.d.) or nonnegative (p.s.d.) eigenvalues. In mathematical literature, kernels are usually complex-valued functions. That is, a complex-valued function K : X × X → C {\displaystyle K:{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {C} } is called a Hermitian kernel if K ( x , y ) = K ( y , x ) ¯ {\displaystyle K(x,y)={\overline {K(y,x)}}} and positive definite if for every finite set of points x 1 , … , x n ∈ X {\displaystyle x_{1},\dots ,x_{n}\in {\mathcal {X}}} and any complex numbers ξ 1 , … , ξ n ∈ C {\displaystyle \xi _{1},\dots ,\xi _{n}\in \mathbb {C} } ,
∑ i = 1 n ∑ j = 1 n ξ i ξ ¯ j K ( x i , x j ) ≥ 0 {\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}\xi _{i}{\overline {\xi }}_{j}K(x_{i},x_{j})\geq 0}
where ξ ¯ j {\displaystyle {\overline {\xi }}_{j}} denotes the complex conjugate. In the rest of this article we assume real-valued functions, which is the common practice in applications of p.d. kernels.
Some general properties For a family of p.d. kernels ( K i ) i ∈ N , K i : X × X → R {\displaystyle (K_{i})_{i\in \mathbb {N} },\ \ K_{i}:{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} }
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