The key idea of the paper is due to Lytvyn and Rvachov (ref. [3]
from 1973) who proposed a replacement of a (truncated) Taylor
series by (truncated) Taylor expansions generated at several
points and matched. Their key trick was a decomposition of a unit
in a finite sum of certain non-negative and smooth functions
$h_m(x)$ which naturally extends to a similar k-term decomposition
of the whole class of functions of a single real variable. The
present authors make a specific ``point-smearing" decomposition
choice of $h_m(x)$ (with $k=2$ for simplicity) and suggest and
describe an application of the trick to the standard (truncated)
continued-fraction approximants. I liked the interesting formulae
for ``reminders" (read: remainders) in the main Theorems 5 and 6.


MR2025760 Kuchmins\cprime ka, Kh. Yo. ; Vozna, S. M. Multipoint
formula based on associated continued fraction. Mat. Metodi
Fiz.-Mekh. Polya 46 (2003), no. 3, 40--47.