Positive-definite kernel

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, embedding problem, information theory, and other areas.Definition: Space H {displaystyle H} is called a reproducing kernel Hilbert space if the evaluation functionals are continuous. Definition: Reproducing kernel is a function K : X × X → R {displaystyle K:X imes X o mathbb {R} } such thatTheorem: Every reproducing kernel K {displaystyle K} induces a unique RKHS, and every RKHS has a unique reproducing kernel. Theorem: Every reproducing kernel is positive definite, and every p.d. kernel defines a unique RKHS, of which it is the unique reproducing kernel. Definition: A symmetric function ψ : X × X → R {displaystyle psi :{mathcal {X}} imes {mathcal {X}} o mathbb {R} } is called a negative definite (n.d.) kernel on X {displaystyle {mathcal {X}}} if 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, embedding problem, information theory, and other areas.