Spectral expansion of the Riemannian heat kernel ($H(t,x,y)={\sum }_j{e}^{-t{\lambda }_j}{\varphi }_j(x)\overline{{\varphi }_j(y)}$)

On a closed manifold with a Riemannian metric, an orthonormal eigenbasis for the positive Laplace-Beltrami operator gives this formula, with eigenvalues counted with multiplicity. Polynomial elliptic bounds and exponential time decay give convergence of every derivative away from time zero. The conjugate is necessary for a complex basis.