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

Abstract

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
$\displaystyle
|f(x)-D_n(x)|\leq \frac{f^*\prod\limits_{k=0}^n |x-x_k|}{(n+1)!\, \Omega_n(t)} \Big( \kappa_{n+1}(p)+\sum_{k=1}^r \tbinom{n+1}{k} (a^*)^k \sum_{i_1=1}^{n+1-2k} \kappa_{i_1}(p)\times$

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

 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

References

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Published
2020-10-05
How to Cite
1.
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
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Articles