Truncation error bounds for a two-dimensional continued $g$-fraction

Kh.Yo.Kuchmins'ka; S.M.Vozna

Truncation error bounds for the two-dimensional continued $g$-fraction have been established in terms of a two-dimensional continued $\pi$-fraction which is an extension of the two-dimensional continued $g$-fraction.

1. Боднар~Д.~И. Ветвящиеся цепные дроби. -- К.: Наук. думка, 1986. -- 176~с.

2. Боднар~Д.~И., Кучминская~Х.~И. Абсолютная сходимость четной и нечетной части двумерной соответствующей цепной дроби // Мат. методы и физ.-мех. поля. -- 1983. Вып. 18. -- С.~30--34.

3. Возна~С.~М. Збіжність двовимірного неперервного $g$-дробу // Мат. методи та фіз.-мех. поля. -- 2004. --- Т.47, № 3. --- C.~28--32.

4. Джоунс~У., Трон~В. Непрерывные дроби. Аналитическая теория и приложения. -- М.: Мир, 1985. -- 414~с.

5. Слешинский~ И.~В. К вопросу о сходимости непрерывных дробей // Мат. сборник. -- 1888. -- Т.14. -- C.~337--343.

6. Bodnar~D.~I., Dmytryshyn~R.~I. On the convergence of multidimensional $g$-fraction // Matematychni Studii. -- 2001. -- V. 15, № 2. -- P. 115--126.

7. Gragg~W.~B. Truncation error bounds for $g$-fractions // Numer. Math. -- 1968. -- V. 11. -- P. 370--379.

8. Kuchmins'ka~Kh. Some properties of two-dimensional continued fractions // J. Comp. and Appl. Math. -- 1999. -- V. 105. -- P.~347--353.

9. Runckel~H. Bounded analytic functions in the unit disk and the behaviour of certain analytic continued fractions near the singular line // J. reine angew. Math. -- 1976. -- V. 281.-- P.~97-125.

10. Thale James~S. Univalence of continued fractions and Stieltjes transform // Proc. Amer. Math. Soc. -- 1956. -- V.7, № 2.-- P.~232-244.

11. Tsygvintsev A.V. On the connection between fractions and solutions of the Feigenbaum-Cvitanovi\'c equation // Communications in the analytic theory of continued fractions. -- 2003. -- V. XI. -- P.~103-112.

12. Wall~H.~S. Analytic theory of continued fractions. -- New York: Van Nostrand, 1948. -- 433~с.

	Truncation error bounds for a two-dimensional continued $g$-fraction
Author	Kh.Yo.Kuchmins'ka, S.M.Vozna khrys1@polynet.lviv.ua National University "Lvivs'ka Politekhnika", Institute of Applied Mathematics and Fundamental Sciences,
Abstract	Truncation error bounds for the two-dimensional continued $g$-fraction have been established in terms of a two-dimensional continued $\pi$-fraction which is an extension of the two-dimensional continued $g$-fraction.
Keywords	truncation error bound, two-dimensional continued fraction, g-fraction, pi-fraction
DOI	doi:10.30970/ms.24.2.120-126
Reference	1. Боднар~Д.~И. Ветвящиеся цепные дроби. -- К.: Наук. думка, 1986. -- 176~с. 2. Боднар~Д.~И., Кучминская~Х.~И. Абсолютная сходимость четной и нечетной части двумерной соответствующей цепной дроби // Мат. методы и физ.-мех. поля. -- 1983. Вып. 18. -- С.~30--34. 3. Возна~С.~М. Збіжність двовимірного неперервного $g$-дробу // Мат. методи та фіз.-мех. поля. -- 2004. --- Т.47, № 3. --- C.~28--32. 4. Джоунс~У., Трон~В. Непрерывные дроби. Аналитическая теория и приложения. -- М.: Мир, 1985. -- 414~с. 5. Слешинский~ И.~В. К вопросу о сходимости непрерывных дробей // Мат. сборник. -- 1888. -- Т.14. -- C.~337--343. 6. Bodnar~D.~I., Dmytryshyn~R.~I. On the convergence of multidimensional $g$-fraction // Matematychni Studii. -- 2001. -- V. 15, № 2. -- P. 115--126. 7. Gragg~W.~B. Truncation error bounds for $g$-fractions // Numer. Math. -- 1968. -- V. 11. -- P. 370--379. 8. Kuchmins'ka~Kh. Some properties of two-dimensional continued fractions // J. Comp. and Appl. Math. -- 1999. -- V. 105. -- P.~347--353. 9. Runckel~H. Bounded analytic functions in the unit disk and the behaviour of certain analytic continued fractions near the singular line // J. reine angew. Math. -- 1976. -- V. 281.-- P.~97-125. 10. Thale James~S. Univalence of continued fractions and Stieltjes transform // Proc. Amer. Math. Soc. -- 1956. -- V.7, № 2.-- P.~232-244. 11. Tsygvintsev A.V. On the connection between fractions and solutions of the Feigenbaum-Cvitanovi\'c equation // Communications in the analytic theory of continued fractions. -- 2003. -- V. XI. -- P.~103-112. 12. Wall~H.~S. Analytic theory of continued fractions. -- New York: Van Nostrand, 1948. -- 433~с.
Pages	120-126
Volume	24
Issue	2
Year	2005
Journal	Matematychni Studii
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