Error Estimates in Interpolation Representation of the Hermite Polynomials and Application for Generalized f-Divergence

Abstract

In this paper, we obtain new estimates for the error function of the Hermitian interpolating polynomial for functions in Cn[a,b]. Specifically, explicit and sharp estimates are derived over the Lp norms (for 1≤p≤∞) of the n-th derivative, taking into account the configuration of the interpolation nodes. As a direct consequence of our main results, we deduce refined error inequalities for the special case of the two-point Taylor interpolation. Furthermore, to demonstrate the utility of these new estimates, we apply our theoretical findings to functionals of Csiszár’s f-divergence type. By applying the basis polynomials of the Hermite interpolation to the generating function f, we derive new inequalities for these functionals. In particular, when f is a convex function, our results provide refined bounds for the classical Csiszár f-divergence. As an application, we obtain new inequalities for the Kullback–Leibler divergence by choosing f(x)=xlnx. This approach successfully connects polynomial approximation theory with information theory.