Multidimensional continued fraction and rational approximation

Abstract

The classical continued fraction is generalized for studying the rational approximation problem on multi-formal Laurent series in this paper, the construction is called m-continued fraction. It is proved that the approximants of an m-continued fraction converge to a multi-formal Laurent series, and are best rational approximations to it; conversely for any multi-formal Laurent series an algorithm called m-CF transform is introduced to obtain its m-continued fraction expansions; moreover, strict m-continued fractions, which are m-continued fractions imposed with some additional conditions, and multi-formal Laurent series are in 1-1 correspondence. It is shown that m-continued fractions can be used to study the multi-sequence synthesis problem.