On the construction of dirichlet series approximations for completely monotone functions

In a series of papers, Liu established and analysed conditions under which completely monotone (CM) functions can be approximated by finite Dirichlet series with positive coefficients. Motivated by a representation theorem of Pollard for Kohlrausch functions, a constructive procedure and proof is given for CM functions which are the Laplace transform of absolutely continuous finite positive measures. The importance of this result, which is new even for Kohlrausch functions, is that it allows accurate approximations to be generated for the Laplace transform of such CM functions which can then be utilized in various ways including the approximate solution of the interconversion relationship of rheology and its generalization for the solution of Volterra integral equations of the first kind.