On an approximation formula

Abstract

We generalize an approximation formula which in some particular cases has been studied by [J.F. Traub 1964] and \ [R.M.Humel and C.S. Secbeck 1949]. Denote by $I_x$ the closed interval determined by the distinct points $x,x_0\in \mathbb{R}$. Consider the nonlinear mapping $f:I_x\to \mathbb{R}$, which has derivatives up to the order $2n+1$ on $I_x$, and deonte by $G$ the set of functions $$G=\{g:g(t)=f(x_0)+(t-x_0)\sum _{i=1}^n{a}_i{f}^{\prime }(x_0+b_i(t-x_0),a_i,b_i\in \mathbb{R},i=1,n,t\in I_x\}$$ From the set $G$ we determine a function $\overline{g}$ with the properties $f^{(i)}(x_0)={\overline{g}}^{(i)}(x_0)$. We determine the coefficients $a_i,b_i,i=1,\dots ,n$ and we also evaluate the remainder $f(t)-\overline{g}(t)$, $t\in I_x$.