The paper investigates the convergence problem of a special class of branched continued fractions, i.e. the multidimensional S-fractions with independent variables, consisting of i1β=1βNβ1ci(1)βzi1βββ+βi2β=1βi1ββ1ci(2)βzi2βββ+βi3β=1βi2ββ1ci(3)βzi3βββ+ββ―, which are multidimensional generalizations of S-fractions (Stieltjes fractions). These branched continued fractions are used, in particular, for approximation of the analytic functions of several variables given by multiple power series. For multidimensional S-fractions with independent variables we have established a convergence criterion in the domain H={z=(z1β,z2β,β¦,zNβ)βCN:β£arg(zkβ+1)β£<Ο,1β€kβ€N} as well as the estimates of the rate of convergence in the open polydisc Q={z=(z1β,z2β,β¦,zNβ)βCN:β£zkββ£<1,1β€kβ€N} and in a closure of the domain $Q.
The starting point of the present paper is a result by E.A. Boltarovych (1989) on convergence regions, dealing with branched continued fraction i1β=1βNβ1ai(1)ββ+βi2β=1βNβ1ai(2)ββ+ββ¦+βinβ=1βNβ1ai(n)ββ+ββ¦, where β£ai(2nβ1)ββ£β€Ξ±/N,ipβ=1,Nβ,p=1,2nβ1β,nβ₯1, and for each multiindex i(2nβ1) there is a single index j2nβ,1β€j2nββ€N, such that β£ai(2nβ1),j2nβββ£β₯R,ipβ=1,Nβ,p=1,2nβ1β,nβ₯1, and β£ai(2n)ββ£β€r/(Nβ1),i2nβξ =j2nβ,ipβ=1,Nβ,p=1,2nβ,nβ₯1, where N>1 and Ξ±,r and R are real numbers that satisfying certain conditions. In the present paper conditions for these regions are replaced by βi1β=1Nββ£ai(1)ββ£β€Ξ±(1βΞ΅),βi2n+1β=1Nββ£ai(2n+1)ββ£β€Ξ±(1βΞ΅),ipβ=1,Nβ,p=1,2nβ,nβ₯1, and for each multiindex i(2nβ1) there is a single index j2nβ,1β€j2nββ€N, such that β£ai(2nβ1),j2nβββ£β₯R and βi2nββ{1,2,β¦,N}\{j2nβ}ββ£ai(2n)ββ£β€r,ipβ=1,Nβ,p=1,2nβ1β,nβ₯1, where Ξ΅,Ξ±,r and R are real numbers that satisfying certain conditions, and better convergence speed estimates are obtained
The paper deals with the problem of approximation of functions of several variables by branched continued fractions. We study the correspondence between formal multiple power series and the so-called "multidimensional S-fraction with independent variables". As a result, the necessary and sufficient conditions for the expansion of the formal multiple power series into the corresponding multidimensional S-fraction with independent variables have been established. Several numerical experiments show the efficiency, power and feasibility of using the branched continued fractions in order to numerically approximate certain functions of several variables from their formal multiple power series