Convergence criterion for branched continued fractions of the special form with positive elements
Journal Title: Карпатські математичні публікації - Year 2017, Vol 9, Issue 1
Abstract
In this paper the problem of convergence of the important type of a multidimensional generalization of continued fractions, the branched continued fractions with independent variables, is considered. This fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. When variables are fixed these fractions are called the branched continued fractions of the special form. Their structure is much simpler then the structure of general branched continued fractions. It has given a possibility to establish the necessary and sufficient conditions of convergence of branched continued fractions of the special form with the positive elements. The received result is the multidimensional analog of Seidel's criterion for the continued fractions. The condition of convergence of investigated fractions is the divergence of series, whose elements are continued fractions. Therefore, the sufficient condition of the convergence of this fraction which has been formulated by the divergence of series composed of partial denominators of this fraction, is established. Using the established criterion and Stieltjes-Vitali Theorem the parabolic theorems of branched continued fractions of the special form with complex elements convergence, is investigated. The sufficient conditions gave a possibility to make the condition of convergence of the branched continued fractions of the special form, whose elements lie in parabolic domains.
Authors and Affiliations
D. Bodnar, I. Bilanyk
Keywords
Related Articles
- EP ID EP324751
- DOI 10.15330/cmp.9.1.13-21
- Views 68
- Downloads 0
On central automorphisms of crossed modules
A crossed module (T,G,∂) consist of a group homomorphism ∂:T→G together with an action (g,t)→gt of G on T satisfying ∂(gt)=g∂(t)g−1 and ∂(s)t=sts−1, for all g∈G and s,t∈T. The term crossed module was introduced by J. H...
Mixed problem for the singular partial differential equation of parabolic type
The scheme for solving of a mixed problem is proposed for a differential equation a(x)∂T∂τ=∂∂x(c(x)∂T∂x)−g(x)T with coefficients a(x), g(x) that are the generalized derivatives of functions of bounded variation, c(x)>0,...
On properties of the solutions of the Weber equation
Growth, convexity and the l-index boundedness of the functions α(z) and β(z), such that α(z4) and zβ(z4) are linear independent solutions of the Weber equation w′′−(z24−ν−12)w=0 if ν=−12 are investigated.
On estimates for the Jacobi transform in the space Lp(R+,Jα,β(x)dx)
In this paper, we prove the estimates for the Jacobi transform in Lp(R+,Jα,β(x)dx) as applied to some classes of functions characterized by a generalized modulus of continuity.
Permanence of a discrete predator-prey system with nonmonotonic functional responses and endless delay
A discrete-time analogue of predator-prey model with nonmonotonic functional responses and endless delay is considered in the paper. We investigate the question of obtaining conditions of permanent behavior of the dynami...