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 \(f:I_{x}\rightarrow \mathbb{R}\), which has derivatives up to the order \(2n+1\) on \(I_{x}\), and deonte by \(G\) the set of functions $$G=\big\{g:g(t) =f(x_0) + (t-x_0) \sum \limits_{i=1}^{n} a_i f'(x_0 + b_i(t-x_0) , \ a_i, b_i \in \mathbb{R}, i=1,n, t\in I_x\big\}$$ From the set \(G\) we determine a function \(\bar{g}\) with the properties \(f^{(i)}(x_0) = \bar{g}^{(i)}(x_0)\). We determine the coefficients \(a_{i},b_{i},\ i=1,\ldots,n\) and we also evaluate the remainder \(f(t) -\bar{g}(t)\), \(t\in I_{x}\).