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.

This article will discuss some of the historical and current developments of the theory of positive definite kernels, starting with the general idea and properties before considering practical applications.

Let ${\displaystyle {\mathcal {X}}}$ be a nonempty set, sometimes referred to as the index set. A symmetric function ${\displaystyle K:{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} }$ is called a positive definite (p.d.) kernel on ${\displaystyle {\mathcal {X}}}$ if

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