Geodesic
Approximation of functions by two-point Hermite interpolating polynomials
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 7, pp. 1091-1108

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A polynomial approximating a given function is constructed assuming that the function and a certain set of its derivatives are known at the endpoints of a given interval. Various analytical formulas are derived for the approximating polynomial. An interpretation of the two-point approximation of the function is given and its relation to the Taylor series expansion of the function is indicated. A sufficient condition for the convergence of a sequence of two-point polynomials to a given function is established. Examples are given in which the sine function is approximated by a sequence of two-point Hermite polynomials on given intervals. The errors in the two-point and Taylor series approximations of the function are compared analytically and numerically. 
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     author = {V. V. Shustov},
     title = {Approximation of functions by two-point {Hermite} interpolating polynomials},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1091--1108},
     publisher = {mathdoc},
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     number = {7},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_7_a0/}
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V. V. Shustov. Approximation of functions by two-point Hermite interpolating polynomials. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 7, pp. 1091-1108. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_7_a0/