The well-known connection between (a) the asymptotic density of eigenvalues of a differential operator H, and (b) the geometry of the region or manifold where H acts, has a local generalization: There is a connection between $({\text{a}'})$ the spectral measures or projection kernel describing the proper normalization, relative to a point $x_0 $, of the expansion of an arbitrary function in eigenfunctions of an operator H (possibly with continuous spectrum), and $({\text{b}'})$ the values of the coefficients (symbol) of H and their derivatives at $x_0 $. Potential applications (especially in general-relativistic quantum field theory) increasingly call for a detailed development of this theory (including calculation of numerical coefficients), which is begun here. The small-time expansion of the Green function of the heat operator, $\frac{\partial }{{\partial t}} + H$, is used to define the “mean” or “effective” expansion of the spectral measures. For the case of one independent and one dependent variable,...