Polynomial approximation of the high orders

A new approach is proposed to high-order polynomial approximation (smoothing), based on the basic elements method (BEM.) The $n^{\mathrm{th}}$-degree BEM-polynomial is expressed using four basic elements given on a three-point grid: $x_0+\alpha<x_0<x_0+\beta$, $\alpha\beta<0$. Formulae have been obtained for calculating the coefficients of the 12-th order polynomial model depending on the interval length, the continuous parameters $\alpha$, $\beta$ and the derivatives $f^{(m)}(x_0+\nu)$, $\nu=\alpha, \beta, 0$, $m=\overline{0,3}$. Application of BEM-polynomials of high degrees for piecewise polynomial approximation (PWA) and smoothing enhances the stability and accuracy of calculations, as the grid step increases, and reduces the computing complexity as well.