Convergence sets and relative stability to perturbations of a branched continued fraction with positive elements

Abstract

In the paper, the problems of convergence and relative stability to perturbations of a branched continued fraction with positive elements and a fixed number of branching branches are investigated. The conditions under which the sets of elements \[\Omega_0 = ( {0,\mu _0^{(2)}} ] \times [ {\nu _0^{(1)}, + \infty } ),\quad \Omega _{i(k)}=[ {\mu _k^{(1)},\mu _k^{(2)}} ] \times [ {\nu _k^{(1)},\nu _k^{(2)}} ],\]\[i(k) \in {I_k}, \quad k = 1,2,\ldots,\] where $\nu _0^{(1)}>0,$ $0 < \mu _k^{(1)} < \mu _k^{(2)},$ $0 < \nu _k^{(1)} < \nu _k^{(2)},$ $k = 1,2,\ldots,$ are a sequence of sets of convergence and relative stability to perturbations of the branched continued fraction \[\frac{a_0}{b_0}{\atop+}\sum_{i_1=1}^N\frac{a_{i(1)}}{b_{i(1)}}{\atop+}\sum_{i_2=1}^N\frac{a_{i(2)}}{b_{i(2)}}{\atop+}\ldots{\atop+} \sum_{i_k=1}^N\frac{a_{i(k)}}{b_{i(k)}}{\atop+}\ldots\] have been established. The obtained conditions require the boundedness or convergence of the sequences whose members depend on the values $\mu _k^{(j)},$ $\nu _k^{(j)},$ $j=1,2.$ If the sets of elements of the branched continued fraction are sets ${\Omega _{i(k)}} = ( {0,{\mu _k}} ] \times [ {{\nu _k}, + \infty } )$, $i(k) \in {I_k}$, $k = 0,1,\ldots,$ where ${\mu _k} > 0$, ${\nu _k} > 0$, $k = 0,1,\ldots,$ then the conditions of convergence and stability to perturbations are formulated through the convergence of series whose terms depend on the values $\mu _k,$ $\nu _k.$ The conditions of relative resistance to perturbations of the branched continued fraction are also established if the partial numerators on the even floors of the fraction are perturbed by a shortage and on the odd ones by an excess, i.e. under the condition that the relative errors of the partial numerators alternate in sign. In all cases, we obtained estimates of the relative errors of the approximants that arise as a result of perturbation of the elements of the branched continued fraction.