Stability to perturbations of continued fraction approximants and applications

The paper investigates the problem of stability to perturbations of continued fraction approximants with complex elements. Unlike the problem of investigating the stability of continued fractions to perturbations, which focuses on the properties of continued fractions, attention is paid to the analysis of the stability to perturbations of their approximants, which have direct practical importance in applied problems. Sufficient conditions for stability to perturbations are obtained in the form of fundamental inequalities for partial numerators. Estimates of relative errors of approximants are proposed, their asymptotic accuracy up to the first order is proved, and the concept of the condition number of the stability problem is introduced. General results are applied to the class of $g$-fractions, and stability sets in ${\mathbb R}$ and ${\mathbb C}$ are constructed. It is shown that the stability sets depend on the fraction parameters. The given examples demonstrate the accuracy of the obtained estimates and their advantages compared to known results.