Representation of a one class function of two variables by bicontinued fractions

M.M. Pahirya

doi:10.15421/241910

Representation of a one class function of two variables by bicontinued fractions

M.M. Pahirya (Uzhhorod National University), https://orcid.org/0000-0003-1488-3302

Abstract

Let function $$$u (z, w) = f (z) h (w)$$$ be defined on the compact set $$$\mathbf{K} \subset \mathbb{C}^2$$$. We study the problem of representation of functions of this class by the product of two continued fractions, which is called a bicontinued fraction. Some properties of Thiele reciprocal derivatives, Thiele continued fractions and regular C-fractions are proved. The possibility of representation of functions of this class by bicontinued fractions is shown. Examples are considered, domains of convergence and uniform convergence of obtained bicontinued fractions to the function are indicated.

Keywords

continued fractions; bicontinued fractions; functions of two complex variables; representation of functions

MSC 2020

30B70; 30E05; 30E10

Full Text:

PDF

References

Korneichuk N.P. Exact constants in approximation theory, Nauka, Moscow, 1987; 424 p. (in Russian)

Dzyadyk V.K., Shevchuk I.A. Theory of uniform approximation of functions by polynomials, Walter de Gruyter, Berlin-New York, 2008; 495 p.

Walsh J.L. Interpolation and approximation by rational functions in complex region, IL, Moscow, 1961; 508 p. (in Russian)

Korneichuk N.P. Splines in approximation theory, Nauka, Moscow, 1984; 544 p. (in Russian)

Henrici P. Applied and computational complex analysis, Vol. 1.: Power series, integration, conformal mapping, location of zeros, John Wiley & Sons, New York-London, 1974; 697 p.

Henrici P. Applied and computational complex analysis, Vol. 2.: Special functions, integral transforms, asymptotics, continued fractions, John Wiley & Sons, New York-London, 1977; 671 p.

Baker (jr) J., Graves-Morris P. Pade approximations, Mir, Moscow, 1986; 502 p. (in Russian)

Khovanskii A.N. The application of continued fractions and their generalizations to problems in approximation theory, P. Noordhoff, Groningen, 1963; 212 p.

Jones W., Thron W. Continued Fractions, Mir, Moscow, 1985; 414 p. (in Russian)

Cuyt A., Brevik P.V., Verdonk B., Waadeland H., Jones W.B. Handbook of continued fractions for special functions, Springer, 2008; 447 p.

Thiele T.N. Interpolationsprechnung, Commisission von B.G. Teubner, Leipzig, 1909; 187 p. (in German)

Nörlund N.E. Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924; 560 p. (in German)

Pahirya M.M. Approximation of functions by continued fractions, Grazhda, Uzhhorod, 2016; 412 p. (in Ukrainian)

Babenko K.I. Introduction to Numerical Analysis, SRC "Regular and chaotic dynamics", Moscow-Izhevsk, 2002; 848 p. (in Russian)

Nikolsky S.M. Approximation of the functions of several variables and inclusion theorems, Nauka, Moscow, 1969; 480 p. (in Russian)

Makarov V.L., Khlobystov V.V., Yanovich L.A. Methods of operator interpolation, Works of Institute of Mathematics, NAS Ukraine, Vol. 83, Kyiv, 2010; 517 p.

Cuyt A. Pade approximations for operators: Theory and applications, Springer-Verlag, Berlin-Heidelberg, 1984; 148 p.

Hildebrand F.B. Introduction to Numerical Analysis, Dover Publications, Inc, New York, 1987; 669 p.

Milne-Thomson L.M. The Calculus of Finite Differences. 2nd ed., AMS Chelsea Publishing, Providence, Rhode Island, 2000; 582 p.

Holub A.P., Chernetz'ka L.O. "Two-dimensional generalized momental representations and rational approximation of functions of two variables", Ukrainian Math. J., 2013, 65(8); pp. 1035-1058. (in Ukrainian) doi:10.1007/s11253-014-0850-x

Holub A.P., Chernetz'ka L.O. "Construction of Pade constants for some hypergeometric Lauricella series by generalized momental representations method", Coll. works of Institute of Mathematics, NAS Ukraine, 2014; 1(1); pp. 63-88. (in Ukrainian)

Skorobohat'ko V.Ya. Theory of branching continued fractions and its application in computational mathematics, Nauka, Moscow, 1983; 312 p. (in Russian)

Kuchmins'ka Kh.J. Two-dimensional continued fractions, Ya.S. Pidstryhach Inst. of appl. problems of mech. and math. of NAS Ukraine, Lviv, 2010; 218 p. (in Ukrainian)

Bodnar D.I. Branched Continued Fractions, Naukova Dumka, Kiev, 1986; 176 p. (in Russian)

Cuyt A., Verdonk B. Different Technique for the Construction of Multivariate Rational Interpolation and Pade Approximants, Universitaire Instelling, Antwerpen, 1988; 158 p.