Mercer kernel ($k(x,z)$)

In the compact-domain spectral setting, a Mercer kernel is a continuous symmetric positive-semidefinite kernel on a compact space equipped with a finite full-support measure. Mercer's theorem yields nonnegative eigenvalues and an expansion $k(x,z)={\sum }_j{\lambda }_j{e}_j(x)e_j(z)$, giving features $(\sqrt{{\lambda }_j}{e}_j(x){)}_j$ in ${\ell }^2$. The finite-Gram positivity condition on arbitrary domains is more general than these spectral hypotheses; all such kernels still have a Reproducing-kernel Hilbert space realization.