Abstract:
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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