Generalised simultaneous approximation of functions

A. J. Van Der Poorten
1991 Journal of the Australian Mathematical Society  
We generalise the approximation theory described in Mahler's paper "Perfect Systems" to linked simultaneous approximations and prove the existence of nonsingular approximation and of transfer matrices by generalising Coates' normality zig-zag theorem. The theory sketched here may have application to constructions important in the theory of diophantine approximation. In 1934, at the University of Groningen, Kurt Mahler gave a course in which he generalised the approximation theory for the
more » ... eory for the exponential function presented in Hermite's fundamental papers [2], [3] . It had seemed that Hermite's theory was unique to the exponential function, but Mahler showed that the theory is quite general and has broad application in the theory of diophantine approximation as well as providing a generalisation of the Padé theory now well known to applied mathematicians cf [5]. Mahler's lectures were not published until many years later [7]. However, in the meantime the manuscript provided a basis for the thesis of Henk Jager [4] and the honours essay (and immediately subsequent work) of John Coates [1]. The present note generalises remarks in [6], which themselves use the ideas of Coates [1] (particularly his notion of normality) as their starting point. The principal innovation of
doi:10.1017/s1446788700033292 fatcat:sg5ohfo7h5f2pllwz2er5zxp6q