Interconversion relationships for completely monotone functions

In linear viscoelasticity, the analytically valid Volterra convolution interconversion relationships between the relaxation modulus G and the corresponding creep compliance (retardation) modulus J play a fundamental role. They allow J to be determined from both theoretical and experimental estimates of G and conversely. In order to guarantee conservation of energy for the related models of linear viscoelastic flow and deformation processes, some assumption such as the complete monotonicity of G and dJ/dt needs to be invoked. Interesting theoretical questions thereby arise about the analytic and structural properties of G and J. Gross [Actualités Sci. Ind. 1190, Hermann, Paris, 1953] appears to have been the first to derive analytical expressions for G in terms of J and conversely. However, the regularity invoked only guarantees their validity for a subset of all possible completely monotone functions. The purpose of this paper is an investigation of the extent to which these results extend to all completely monotone functions. This allows issues associated with the effect of perturbations in G on J, and conversely, to be placed on a rigorous footing. In particular, it is shown, among other things, that G (resp., dJ/dt) having absolutely continuous generating measure does not necessarily guarantee the same for dJ/dt (resp., G). Our results have been derived by using the equivalent resolvent kernel equation that comes from a double differentiation of the interconversion equation. Consequently, they will hold more generally for resolvent kernel equations.