On the convergence of multidimensional S-fractions with independent variables
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}\f...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2020-12-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
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| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/4320 |
| _version_ | 1827281260440977408 |
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| author | O.S. Bodnar R.I. Dmytryshyn S.V. Sharyn |
| author_facet | O.S. Bodnar R.I. Dmytryshyn S.V. Sharyn |
| author_sort | O.S. Bodnar |
| collection | DOAJ |
| description | 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 $Q.$ |
| first_indexed | 2024-04-24T08:57:15Z |
| format | Article |
| id | doaj.art-2150c25458ef4d0da23dd020c80b756a |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:57:15Z |
| publishDate | 2020-12-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-2150c25458ef4d0da23dd020c80b756a2024-04-16T07:04:01ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102020-12-0112235335910.15330/cmp.12.2.353-3593783On the convergence of multidimensional S-fractions with independent variablesO.S. Bodnar0R.I. Dmytryshyn1https://orcid.org/0000-0003-2845-0137S.V. Sharyn2https://orcid.org/0000-0003-2547-1442Volodymyr Gnatiuk Ternopil National Pedagogical University, 2 Kryvonosa str., 46027, Ternopil, UkraineVasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, UkraineVasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, UkraineThe 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 $Q.$https://journals.pnu.edu.ua/index.php/cmp/article/view/4320branched continued fractionconvergence criterionuniform convergenceestimates of the rate of convergencecontinued fraction |
| spellingShingle | O.S. Bodnar R.I. Dmytryshyn S.V. Sharyn On the convergence of multidimensional S-fractions with independent variables Karpatsʹkì Matematičnì Publìkacìï branched continued fraction convergence criterion uniform convergence estimates of the rate of convergence continued fraction |
| title | On the convergence of multidimensional S-fractions with independent variables |
| title_full | On the convergence of multidimensional S-fractions with independent variables |
| title_fullStr | On the convergence of multidimensional S-fractions with independent variables |
| title_full_unstemmed | On the convergence of multidimensional S-fractions with independent variables |
| title_short | On the convergence of multidimensional S-fractions with independent variables |
| title_sort | on the convergence of multidimensional s fractions with independent variables |
| topic | branched continued fraction convergence criterion uniform convergence estimates of the rate of convergence continued fraction |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/4320 |
| work_keys_str_mv | AT osbodnar ontheconvergenceofmultidimensionalsfractionswithindependentvariables AT ridmytryshyn ontheconvergenceofmultidimensionalsfractionswithindependentvariables AT svsharyn ontheconvergenceofmultidimensionalsfractionswithindependentvariables |