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

Анотація

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].
$