Rigorous Bounds for Time-Dependent Correlation Functions

Rigorous upper and lower bounds are determined for time-dependent correlation functions, of the type used in statistical mechanics and spectroscopy. The input data are the values of any finite number of initial time derivatives of the correlation function. As an example, bounds are found for the classical velocity correlation function for a lattice vibration problem. The bounds are found to be much more accurate than the Taylor series based on the same time derivatives.