Definition and uniqueness of integral approximants

摘要

Definitions are given for the integral approximant polynomials which insure their existence and uniqueness. A specification of minimality is required in these definitions. Existence of infinite subsequences of integral polynomials without common factors of z is proven. An equivalence theorem between the property of agreement of all high-order integral polynomials and the property that the function being approximated belongs to a particular function class is proved. The accuracy-through-order property is found to hold for all the cases we have investigated for the integral approximant. An example is given which proves that the series coefficients which uniquely determine the integral polynomials may not uniquely determine the integral approximant.