Padé approximants to physical properties via inner projections

The [N, M] Padé approximants to functions formally associated to power series expanssions are expressed in terms of expectation values of inverse matrices. These formulae, which can be derived by the inner‐projection technique, lead to a simple analysis of the properties of serveral approximation methods and their inter‐relationships, in particular Gaussian integration, continued factorization and Padé approximations, which are of current interest in the calculation of physical properties. A relation with Fredholm integral equations and expansions of the resolvent is also discussed. The use of operator inequalities in a systematic fashion is particularly convenient when both the function being approximated and the coefficients of the power series have physically meaningful expressions as moments of operators.