Karhunen-Loève Eigenmode Expansion

The Karhunen-Loève (KL) theorem provides the optimal decomposition of a random field f(x) into orthonormal eigenmodes φₖ of the covariance kernel C(x,y) = E[f(x)f(y)]. Result: f(x) = Σₖ √λₖ ξₖ φₖ(x) where ξₖ ~ N(0,1) are uncorrelated random coefficients. This is the minimum mean-square error representation for any truncated expansion — modes ordered by variance captured (eigenvalue λₖ). Used in PCA, dimensionality reduction, weather modeling, and turbulence analysis.