Partial differential approximants for multivariable power series. I. Definitions and faithfulness

Abstract

The approximation of functions of two (or more) variables given in terms of their power series is discussed. Partial differential approximants, and their multipoint and constrained extensions, are defined systematically and their theory is developed. The questions of existence and uniqueness are addressed and broad criteria are established under which the approxi­mants are faithful in that their expansions match the originally given series to appropriate order. The problem is raised of identifying classes of approximants possessing desirable invariance properties: part II (in preparation) attacks this generally and provides explicit criteria for invariance under Euler transformations and under homogeneous linear transformations of the variables.