scholarly journals Approximation of functions of several variables by multidimensional $S$-fractions with independent variables

2021 ◽  
Vol 13 (3) ◽  
pp. 592-607
Author(s):  
R.I. Dmytryshyn ◽  
S.V. Sharyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Numerical Experiments ◽  
Sufficient Conditions ◽  
Approximation Of Functions ◽  
Independent Variables ◽  
Several Variables ◽  
Multiple Power ◽  
Necessary And Sufficient ◽  
Multiple Power Series

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.

Multidimensional regular C-fraction with independent variables corresponding to formal multiple power series

2019 ◽  
Vol 150 (4) ◽  
pp. 1853-1870 ◽  
Author(s):  
R. I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Numerical Experiments ◽  
The Other ◽  
Independent Variables ◽  
Classical Algorithm ◽  
Multiple Power ◽  
The One ◽  
The Given ◽  
Multiple Power Series

AbstractIn the paper the correspondence between a formal multiple power series and a special type of branched continued fractions, the so-called ‘multidimensional regular C-fractions with independent variables’ is analysed providing with an algorithm based upon the classical algorithm and that enables us to compute from the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional regular C-fraction with independent variables. A few numerical experiments show, on the one hand, the efficiency of the proposed algorithm and, on the other, the power and feasibility of the method in order to numerically approximate certain multivariable functions from their formal multiple power series.

scholarly journals Convergence criterion for branched contіnued fractions of the special form with positive elements

2017 ◽  
Vol 9 (1) ◽  
pp. 13-21 ◽  
Author(s):  
D.I. Bodnar ◽  
I.B. Bilanyk
Keyword(s):  
Continued Fractions ◽  
Special Form ◽  
Sufficient Conditions ◽  
Convergence Criterion ◽  
Sufficient Condition ◽  
Independent Variables ◽  
Positive Elements ◽  
Multiple Power ◽  
Necessary And Sufficient ◽  
Multiple Power Series

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.

scholarly journals On the convergence of multidimensional S-fractions with independent variables

2020 ◽  
Vol 12 (2) ◽  
pp. 353-359
Author(s):  
O.S. Bodnar ◽  
R.I. Dmytryshyn ◽  
S.V. Sharyn
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Continued Fractions ◽  
Special Class ◽  
Convergence Criterion ◽  
Convergence Problem ◽  
Independent Variables ◽  
Several Variables ◽  
Multiple Power ◽  
Multiple Power Series

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 \[\sum_{i_1=1}^N\frac{c_{i(1)}z_{i_1}}{1}{\atop+}\sum_{i_2=1}^{i_1}\frac{c_{i(2)}z_{i_2}}{1}{\atop+} \sum_{i_3=1}^{i_2}\frac{c_{i(3)}z_{i_3}}{1}{\atop+}\cdots,\] 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 \[H=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|\arg(z_k+1)|<\pi,\; 1\le k\le N\right\}\] as well as the estimates of the rate of convergence in the open polydisc \[Q=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|z_k|<1,\;1\le k\le N\right\}\] and in a closure of the domain $Q.$

scholarly journals A multidimensional generalization of the Rutishauser $qd$-algorithm

2016 ◽  
Vol 8 (2) ◽  
pp. 230-238 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Multiple Power ◽  
Necessary And Sufficient ◽  
Multiple Power Series

In this paper the regular multidimensional $C$-fraction with independent variables, which is a generalization of regular $C$-fraction, is considered. An algorithm of calculation of the coefficients of the regular multidimensional $C$-fraction with independent variables correspondence to a given formal multiple power series is constructed. Necessary and sufficient conditions of the existence of this algorithm are established. The above mentioned algorithm is a multidimensional generalization of the Rutishauser $qd$-algorithm.

scholarly journals On the convergence of multidimensional regular C-fractions with independent variables

2018 ◽  
Vol 26 (1) ◽  
pp. 18 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Open Disk ◽  
Efficient Tool ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Multiple Power ◽  
Multiple Power Series ◽  
Branched Continued Fraction

In this paper, we investigate the convergence of multidimensional regular С-fractions with independent variables, which are a multidimensional generalization of regular С-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. We have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional regular С-fraction with independent variables. And, in addition, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional regular С-fraction with independent variables.

scholarly journals On the convergence criterion for branched continued fractions with independent variables

2018 ◽  
Vol 9 (2) ◽  
pp. 120-127 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Special Form ◽  
Absolute Convergence ◽  
Convergence Criterion ◽  
Complex Numbers ◽  
Independent Variables ◽  
Nonnegative Coefficients ◽  
Multiple Power ◽  
Multiple Power Series

In this paper, we consider the problem of convergence of an important type of multidimensional generalization of continued fractions, the branched continued fractions with independent variables. These fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. We have established the effective criterion of absolute convergence of branched continued fractions of the special form in the case when the partial numerators are complex numbers and partial denominators are equal to one. This result is a multidimensional analog of the Worpitzky's criterion for continued fractions. We have investigated the polycircular domain of uniform convergence for multidimensional C-fractions with independent variables in the case of nonnegative coefficients of this fraction.

scholarly journals Local Limit Theorem for the Multiple Power Series Distributions

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2067
Author(s):  
Arsen L. Yakymiv
Keyword(s):  
Limit Theorem ◽  
Power Series ◽  
Local Limit Theorem ◽  
Sufficient Conditions ◽  
Local Limit ◽  
Necessary And Sufficient Conditions ◽  
Multiple Power ◽  
Power Series Distributions ◽  
Necessary And Sufficient ◽  
Multiple Power Series

We study the behavior of multiple power series distributions at the boundary points of their existence. In previous papers, the necessary and sufficient conditions for the integral limit theorem were obtained. Here, the necessary and sufficient conditions for the corresponding local limit theorem are established. This article is dedicated to the memory of my teacher, professor V.M. Zolotarev.

scholarly journals On the convergence of multidimensional S-fractions with independent variables

2018 ◽  
Vol 10 (1) ◽  
pp. 58-64
Author(s):  
O.S. Bodnar ◽  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fraction ◽  
Small Region ◽  
Open Disk ◽  
Convergence Criteria ◽  
Efficient Tool ◽  
Independent Variables ◽  
Multiple Power ◽  
Multiple Power Series ◽  
Branched Continued Fraction

In this paper, we investigate the convergence of multidimensional S-fractions with independent variables, which are a multidimensional generalization of S-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. For establishing the convergence criteria, we use the convergence continuation theorem to extend the convergence, already known for a small region, to a larger region. As a result, we have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional S-fraction with independent variables. And, also, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional S-fraction with independent variables. In addition, we have obtained two new convergence criteria for S-fractions as a consequences from the above mentioned results.

scholarly journals Some properties of approximants for branched continued fractions of the special form with positive and alternating-sign partial numerators

2018 ◽  
Vol 10 (1) ◽  
pp. 3-13 ◽  
Author(s):  
T.M. Antonova ◽  
M.V. Dmytryshyn ◽  
S.M. Vozna
Keyword(s):  
Continued Fractions ◽  
Special Form ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Rational Approximants ◽  
Independent Variables ◽  
Function Of Two Variables ◽  
Necessary And Sufficient ◽  
Double Power Series ◽  
Two Independent Variables

The paper deals with research of convergence for one of the generalizations of continued fractions -- branched continued fractions of the special form with two branches. Such branched continued fractions, similarly as the two-dimensional continued fractions and the branched continued fractions with two independent variables are connected with the problem of  the correspondence between a formal double power series and a sequence of the rational approximants of a function of two variables. Unlike continued fractions, approximants of which are constructed unambiguously, there are many ways to construct approximants of branched continued fractions of the general and the special form. The paper examines the ordinary approximants and one of the structures of figured approximants of the studied branched continued fractions, which is connected with the problem of correspondence. We consider some properties of approximants of such fractions, whose partial numerators are positive and alternating-sign  and partial denominators are equal to one. Some necessary and sufficient conditions for figured convergence are established. It is proved that under these conditions from the convergence of the sequence of figured approximants it follows the convergence of the sequence of ordinary approximants  to the same limit.

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