Concept

Mercer's theorem

Summary
In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in , is one of the most notable results of the work of James Mercer (1883–1932). It is an important theoretical tool in the theory of integral equations; it is used in the Hilbert space theory of stochastic processes, for example the Karhunen–Loève theorem; and it is also used to characterize a symmetric positive-definite kernel. Introduction To explain Mercer's theorem, we first consider an important special case; see below for a more general formulation. A kernel, in this context, is a symmetric continuous function : K: [a,b] \times [a,b] \rightarrow \mathbb{R} where symmetric means that K(x,y) = K(y,x) for all x,y \in [a,b]. K is said to be positive-definite if and only if : \sum_{i=1}^n\sum_{j=1}^n K(x_i
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading