MercerSatz

MercerSatz, commonly known as Mercer’s theorem, is a result in functional analysis and integral equations named after James Mercer. It provides a precise representation of continuous, symmetric, positive semidefinite kernels on compact spaces as a uniformly convergent series of eigenfunctions.

The theorem considers a compact domain X with a finite measure and a kernel K: X × X → R that is continuous, symmetric (K(x,y) = K(y,x)), and positive semidefinite in the sense that for any square-integrable function f on X, the double integral ∫∫ K(x,y) f(x) f(y) dμ(x) dμ(y) is nonnegative. Under these conditions there exist nonnegative eigenvalues λ_i and a corresponding orthonormal system of eigenfunctions φ_i in L^2(μ) such that, for all x and y in X, K(x,y) = sum_i λ_i φ_i(x) φ_i(y). The convergence of this series is uniform on X × X, and if K is continuous on a compact domain, the φ_i can be chosen to be continuous as well.

Consequences of Mercer’s theorem include the interpretation of K as the kernel of a compact, self-adjoint, positive operator T_K on L^2(μ), with the eigenfunctions φ_i forming an orthonormal basis. The kernel can be approximated by finite-rank representations using partial sums of the series. The theorem underpins many practical methods, notably in statistics and machine learning through kernel methods such as support vector machines and Gaussian processes, as well as in the numerical solution of integral equations.

See also: positive definite kernels, Mercer kernels, spectral theory.