Note on Taylor's formula and some applications (Q1099348)

Let \(w_ i\) \((i=0,1,...,n)\) be functions of class \({\mathcal C}^{n- i}[a,b]\), either positive or negative on [a,b], and let \(D_ j\), \(j=0,1,...,n\), denote the first-order differential operator \((D_ jf)(t)=d/dt(f(t)/w_ j(t)).\) The authors prove a generalization of Taylor's formula, namely that a real function, f: [a,b]\(\to {\mathbb{R}}\) such that \((D_ nD_{n-1}...D_ 0f)(x)\) is continuous on [a,b] is representable in the form \[ f(t)=\sum^{n}_{i=0}a_ iW_ i(t_ ic)+R_ n(t),\quad \forall t\in [a,b], \] where \(c\in [a,b]\), \(W_ k\) is expressed in form of integrals of \(w_ 0,...,w_ k\) and \(R_ n\) is a corresponding remainder term. This result is applied to extended complete Chebyshev systems, to ordinary Taylor series and to some estimates for the function \(\int^{x}_{0}\exp (t\) 2)dt.