On convergence and estimation of the truncation error of corresponding two-dimensional continued fractions

  • T. M. Antonova Lviv Polytechnic National University
  • O. M. Sus' Ya. S. Pіdstryhach Institute for Applied Problems of Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv
  • S. M. Vozna Lviv Polytechnic National University

Abstract

UDC 517.524

For the corresponding two-dimensional continued fractions with complex partial numerators belonging to some subsets of the Cartesian product of two angular sets in the right half-plane and partial denominators equal to one, we establish sufficient conditions of uniform convergence and an estimation of the truncation error using an analogue of the method of fundamental inequalities, formulas for real and imagine parts of tails of figured approximants and a multidimensional analogue of the Stieltjes-Vitali theorem.


Author Biographies

O. M. Sus', Ya. S. Pіdstryhach Institute for Applied Problems of Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv



S. M. Vozna, Lviv Polytechnic National University

 

 

 

References

T. M. Antonova, Shvidkist' zbizhnosti gillyastih lancyugovih drobiv special'nogo viglyadu, Volin. mat. visn., 6, 5 – 11 (1999).

T. M. Antonova, D. I. Bodnar, Oblasti zbizhnosti gillyastih lancyugovih drobiv special'nogo viglyadu, Teoriya nablizhennya funkcij ta її zastosuvannya, Praci In-tu matematiki NAN Ukraїni,31, 5 – 18 (2000).

T. M. Antonova, S. M. Vozna, Pro odin analog metodu fundamental'nih nerivnostej dlya doslidzhennya zbizhnosti gillyastih lancyugovih drobiv special'nogo viglyadu, Visn. Nac. un-tu „L'viv. politekhnika”, Ser. fiz.- mat. nauk, 871, 5 – 12 (2017).

T. M. Antonova, S. M. Vozna, Pro zbizhnist' odnogo klasu dvovimirnih vidpovidnih gillyastih lancyugovih drobiv, Prikl. probl. mekhaniki i matematiki, vip.18, 25 – 33 (2020). DOI: https://doi.org/10.15407/apmm2020.18.25-33

T. M. Antonova, R. I. Dmitrishin, Ocinki pohibki nablizhennya dlya gillyastogo lancyugovogo drobu $ Ʃ^N_{i_1 = 1} ai(1) /1 + Ʃ^{i_1}_{i_2 = 1} ai(2) /1 + Ʃ^{i_2}_{i_3 = 1} ai(3)/1 + . . . $, Ukr. mat. zhurn.,72, № 7, 877 – 885 (2020).

T. M. Antonova, O. M. Sus', Pro parni mnozhini zbizhnosti dlya dvovimirnih neperervnih drobiv iz kompleksnimi elementami, Mat. metodi ta fiz.-mekh. polya,50, № 3, 94 – 101 (2007).

T. M. Antonova, O. M. Sus', Pro odnu oznaku figurnoї zbizhnosti dvovimirnih neperervnih drobiv z kompleksnimi elementami, Mat. metodi ta fiz.-mekh. polya,52, № 2, 28 – 35 (2009).

T. M. Antonova, O. M. Sus', Pro deyaki poslidovnosti mnozhin rivnomirnoї zbizhnosti dvovimirnih neperervnih drobiv, Mat. metodi ta fiz.-mekh. polya,58, № 1, 47 – 56 (2015).

D. I. Bodnar, Vetvyashchiesya cepnye drobi, Nauk. dumka, Kiїv(1986).

D. I. Bodnar, Bilanik I. B .,Pro zbizhnist' gillyastih lancyugovih drobiv u kutovih oblastyah, Mat. metodi ta fiz.-mekh. polya,60, № 3, 60 – 69 (2017).

D. I. Bodnar, I. B. Bilanik, Ocinki shvidkosti potochkovoї ta rivnomirnoї zbizhnosti GLD z nerivnoznachnimi zminnimi, Mat. metodi ta fiz.-mekh. polya,62, № 4, 72 – 82 (2019).

D. I. Bodnar, R. I. Dmitrishin, Bagatovimirni priєdnani drobi z nerivnoznachnimi zminnimi i kratni stepenevi ryadi, Ukr. mat. zhurn.,71, № 3, 325 – 339 (2019).

S. M. Vozna, Pro zbizhnist' dvovimirnogo neperervnogo $g$-drobu, Mat. metodi ta fiz.-mekh. polya,47, № 3, 28 – 32 (2004).

С. M. Vozna, H. J. Kuchmins'ka, Vidpovidnist' mizh formal'nim podvijnim stepenevim ryadom i dvovimirnim neperervnim $g$-drobom, Problemi teoriї nablizhennya funkcij ta sumizhni pitannya, Zbirnik prac' In-tu matematiki NAN Ukraїni,1, № 4, 130 – 142 (2004).

R. I. Dmitrishin, Pro rozvinennya deyakih funkcij u dvovimirnij g-drib z nerivnoznachnimi zminnimi, Mat. metodi ta fiz.-mekh. polya,53, № 4, 28 – 34 (2010).

R. I. Dmitrishin, Dvovimirne uzagal'nennya $qd$-algoritmu Rutiskhauzera, Mat. metodi ta fiz.-mekh. polya, 56, № 4, 6 – 11 (2013).

R. I. Dmitrishin, Priєdnani gillyasti lancyugovi drobi z dvoma nerivnoznachnimi zminnimi, Ukr. mat. zhurn., 66, № 9, 1175 – 1184 (2014).

H. J. Kuchmins'ka, Vidpovidnij i priєdnanij gillyasti lancyugovi drobi dlya podvijnogo stepenevogo ryadu, Dop. AH URSR. Ser. А, № 7, 614 – 618 (1978).

H. J. Kuchmins'ka, Dvovimirni neperervni drobi, In-t prikl. probl. mekhaniki i matematiki NAN Ukraїni, L'viv (2010).

H. J. Kuchmins'ka, O. M. Sus', S. M. Vozna, Aproksimativni vlastivosti dvovimirnih neperervnih drobiv, Ukr. mat. zhurn., 55, № 1, 30 – 44 (2003). DOI: https://doi.org/10.1023/A:1025016501397

O. M. Sus', Pro odin iz analogiv metodu fundamental'nih nerivnostej dlya dvovimirnih neperervnih drobiv, Prikl. probl. mekhaniki i matematiki, vip. 5, 71 – 76 (2007).

T. M. Antonova, R. I. Dmytryshyn, Truncation error bounds for branched continued fraction whose partial denominators are equal to unity, Mat. Stud., 54, № 1, 3 – 14 (2020), https://doi.org/10.30970/ms.54.1.3-14 DOI: https://doi.org/10.30970/ms.54.1.3-14

R. I. Dmytryshyn, The two-dimensional $g$-fraction with independent variables for double power series, J. Approx. Theory, 164, № 12, 1520 – 1539 (2012), https://doi.org/10.1016/j.jat.2012.09.002 DOI: https://doi.org/10.1016/j.jat.2012.09.002

R. I. Dmytryshyn, Multidimensional regular $C$ - fraction with independent variables corresponding to formal multiple power series, Proc. Roy. Soc. Edinburgh Sect. A, 1 – 18 (2019); https://doi.org/10.1017/prm.2019.2. DOI: https://doi.org/10.1017/prm.2019.2

R. I. Dmytryshyn, On some of convergence domains of multidimensional $S$-fractions with independent variables, Carpathian Math. Publ., 11, № 1, 54 – 58 (2019), https://doi.org/10.15421/241803 DOI: https://doi.org/10.15330/cmp.11.1.54-58

R. I. Dmytryshyn, S. V. Sharyn, Approximation of functions of several variables by multidimensional $S$-fractions with independent variables, Carpathian Math. Publ., 13, № 3, 592 – 607 (2021), https://doi.org/10.15330/cmp.13.3.592-607 DOI: https://doi.org/10.15330/cmp.13.3.592-607

W. B. Jones, W. J. Thron, Continued fractions: analytic theory and applications, Addison-Wesley Pub. Co., Reading, Mas. (1980).

Kh. Yo. Kuchmins’ka, S. M. Vozna, Truncation error bounds for a two-dimensional continued $g$-fraction, Mat. Stud., 24, № 2, 120 – 126 (2005).

J. Murphy, M. R. O’Donohoe, A two-variable generalization of the Stieltjes-type continued fractions, J. Comput. and Appl. Math., 4, № 3, 181 – 190 (1978), https://doi.org/10.1016/0771-050X(78)90002-5 DOI: https://doi.org/10.1016/0771-050X(78)90002-5

W. Siemaszko, Branched continued fraction for double power series, J. Comp. and Appl. Math., 6, № 2, 121 – 125 (1980), https://doi.org/10.1016/0771-050X(80)90005-4 DOI: https://doi.org/10.1016/0771-050X(80)90005-4

Published
19.05.2022
How to Cite
AntonovaT. M., Sus’O. M., and VoznaS. M. “On Convergence and Estimation of the Truncation Error of Corresponding Two-Dimensional Continued Fractions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 4, May 2022, pp. 443 -57, doi:10.37863/umzh.v74i4.7031.
Section
Research articles