Weakly perfect systems of functions and multidimensional continued fractions
The author continues his work on expansions into continued fractions by the Jacobi-Perron algorithm of m-dimensional vectors \(\bar a\), \(\bar a=(a_ 1,...,a_ m)\) whose components are power series about \(z=\infty\) with finite principal part: \(a_ i=\sum^{\infty}_{j=-\ell_ i}e_{i,j}z^{-j},\quad e_{i,j}\in {\mathbb{C}},\quad\ell_ i\in {\mathbb{Z}}.\) \textit{E. M. Nikishin} [Mat. Sb., Nov. Ser. 113(155), 499-519 (1980; Zbl 0456.30009)] introduced a system of weakly perfect functions. In this paper a connection between this concept and the so-called regular n-dimensional continued fraction is established. The following theorem is proved. In order that the system of series \(a_ i\), \(i=1,...,m\), above be weakly perfect it is necessary and sufficient that a vector \(\bar a\) above may be expanded in an infinite regular continued fraction \(0+\frac{1}{\bar p_ 1+}\frac{1}{\bar p_ 2+...}.\)
scientific article; zbMATH DE number 3869674