Mercerexpansion

Mercer expansion refers to a representation of a continuous symmetric positive-definite kernel as a sum of eigenfunctions of an associated integral operator. It is a consequence of Mercer's theorem, which applies to kernels defined on a compact domain with respect to a finite measure.

Setup and statement. Let Ω be a compact domain and K: Ω×Ω → R be continuous, symmetric (K(x,y) = K(y,x)) and positive semi-definite. Define the integral operator T_K by (T_K f)(x) = ∫_Ω K(x,y) f(y) dμ(y), where μ is a finite measure on Ω. Then T_K is compact and self-adjoint on L2(Ω, μ). There exist nonnegative eigenvalues {λ_i} and corresponding orthonormal eigenfunctions {φ_i} in L2(Ω, μ) such that T_K φ_i = λ_i φ_i.

Mercer expansion. The kernel K admits the expansion K(x,y) = ∑_{i=1}^∞ λ_i φ_i(x) φ_i(y) for all x,y ∈ Ω, with uniform convergence on Ω×Ω. The eigenfunctions {φ_i} form an orthonormal basis in L2(Ω, μ), and the sequence {λ_i} is nonincreasing with λ_i ≥ 0 and λ_i → 0 as i → ∞.

Interpretation and uses. The expansion expresses K as an inner product in a reproducing feature space: K(x,y) = ⟨Φ(x), Φ(y)⟩ where Φ(x) = [√λ_i φ_i(x)]. This yields low-rank or finite-rank approximations by truncating the series. Mercer expansions underpin kernel methods in machine learning, such as kernel PCA and Gaussian processes, and enable practical Nyström-type discretizations for large-scale problems.

Limitations. The theorem requires a compact domain, continuity, and positive definiteness. Non-compact domains or non-symmetric kernels may not admit a Mercer expansion.