A continuant and an estimate of the remainder of the interpolating continued C-fraction

  • M.M. Pahirya Mukachevo State University
Keywords: continued fraction; continuant; intrpolation function; estimate of the remainder


The problem of the interpolation of functions of a real variable by interpolating continued $C$-fraction is investigated. The relationship between the continued fraction and the continuant was used. The properties of the continuant are established. The formula for the remainder of the interpolating continued $C$-fraction proved. The remainder expressed in terms of derivatives of the functional continent. An estimate of the remainder was obtained. The main result of this paper is contained in the following Theorem 5:

Let \(\mathcal{R}\subset \mathbb{R} \) be a compact, \(f \in \mathbf{C}^{(n+1)}(\mathcal{R})\) and
the interpolating continued $C$-fraction~($C$-ICF) of the form
D_n(x)=\frac{P_n(x)}{Q_n(x)}=a_0+\bfrac{K}{k=1}{n}\frac{a_k(x-x_{k-1})}{1}, \ a_k \in \mathbb{R}, \; k=\overline{0,n},$$
be constructed by the values the function \(f\) at nodes $
X=\{x_i : x_i \in \mathcal{R}, x_i\neq x_j, i\neq j, i,j=\overline{0,n}\}.$
If the partial numerators of $C$-ICF satisfy the condition of the Paydon--Wall type, that is
\(0<a^* \ {\rm diam}\, \mathcal{R} \leq p\), then
|f(x)-D_n(x)|\leq \frac{f^*\prod\limits_{k=0}^n |x-x_k|}{(n+1)!\, \Omega_n(t)} \Big( \kappa_{n+1}(p)+\sum_{k=1}^r \tbinom{n+1}{k} (a^*)^k \sum_{i_1=1}^{n+1-2k} \kappa_{i_1}(p)\times$

\times \sum_{i_2=i_1+2}^{n+3-3k} \kappa_{i_2-i_1-1}(p)\dots
\sum_{i_{k-1}=i_{k-2}+2}^{n-3} \kappa_{i_{k-1}-i_{k-2}-1}(p)
\sum_{i_k=i_{k-1}+2}^{n-1} \kappa_{i_k-i_{k-1}-1}(p)
\, \kappa_{n-i_{k}}(p)\Big),$

 where $\displaystyle f^*=
\max\limits_{0\leq m \leq r}\max\limits_{x \in \mathcal{R}} |f^{(n+1-m)}(x)|,$
$\displaystyle \kappa_n(p)=\cfrac{(1\!+\!\sqrt{1+4p})^n\!-\!(1\!-\!\sqrt{1+4p})^n}{2^n\, \sqrt{1+4p}},$\
$a^*=\max\limits_{2\leqslant i \leqslant n}|a_i|,$\ $p=t(1-t),\;
t\in(0;\tfrac{1}{2}], \; r=\big[\tfrac{n}{2}\big].

Author Biography

M.M. Pahirya, Mukachevo State University

Uzhhorod National University, Mukachevo State University, Ukraine


I.P Gavrilyuk, V.L. Makarov, Calculation methods, Kyiv: Vysha Shcola, 1995, 367 p. (in Ukrainian)

A.A. Privalov, Theory of the interpolation of functions, Saratov: Izdatel’stvo Saratovskogo Universiteta, 1990, 424 p. (in Russian)

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

V.L. Makarov, V.V. Khlobystov, Spline approximation of functions, Moskva: Vyshaya Shcola, 1983, 80 p. (in Russian)

G.A. Baker, P. Graves–Morris, Pad´e Approximants, Addison–Wesley Publ. Comp., 1981, 540 p.

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

M.M. Pahirya, Estimation of the remainder for the interpolation continued C-fraction, Ukr. Mat. Zh., 66 (2014), №6, 806–814, (in Ukrainian); Engl. trasl. Ukr. Math. J., 66 (2014), №6, 905–915.

W.B. Jones, W.J. Thron, Continued fractions. Analytic theory and applications, Addison–Wesley Publ. Comp., 1980, 428 p.

T.N. Thiele, Interpolationsprechnung, Leipzig: Commisission von B.G. Teubner, 1909, XII+175S.

G. Chrystal, Algebra, Vol. II, London, A.C. Black, 1889, XXIV+616 p.

R. Vein, P. Dale, Determinants and their application in mathematical physics, Springer Science & Business Media, 2006, VIII+376 p.

A.G. Kurosh, A course in higher algebra, Мoscow.: Mir, 1988, 432 p. (in Russian)

G.M. Fihtengol’c, A course in differential and integral calculus. V.I., Moskva, Nauka, 2003, 680 p. (in Russian)

K.I. Babenko, Foundations of numerical analysis, Moskva–Izhevsk: Regularnaya & Haoticheskaya dynamika, 2002, 848 p. (in Russian)

How to Cite
Pahirya M. A continuant and an estimate of the remainder of the interpolating continued C-fraction . Mat. Stud. [Internet]. 2020Oct.5 [cited 2020Oct.27];54(1):32-5. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/21