N-VARIABLE RATIONAL APPROXIMANTS

Abstract

“Desirable properties” of a two-variable generalization of Pade approximants are laid down. The “Chisholm approximants” are defined and are shown to obey nearly all of these properties; the alternative ways of completing a unique definition are discussed, and the “prong structure” of the defining equations is elucidated. Several generalizations and variants of Chisholm approximants are described: N-variable diagonal (Chisholm and McEwan), 2-variable simple off-diagonal (Graves-Morris, Hughes Jones and Makinson), N-variable simple and general off-diagonal (Hughes Jones), and rotationally covariant 2-variable approximants (Chisholm and Roberts). All of the 2-variable approximants are capable of representing singularities of functions of two variables, and of analytically continuing beyond the polycylinder of convergence of the double series.