Hermite-type interpolation in terms of exponential polynomials

TL;DR

This paper introduces an Hermite-type interpolation method using exponential polynomials, providing an integral error estimate for cases involving differential operators with constant coefficients.

Contribution

It develops a new Hermite-type interpolation framework based on exponential polynomials and derives an integral form error estimate for kernels of linear differential operators.

Findings

Error term expressed in integral form for the interpolation

Applicability demonstrated through several corollaries

Extension of classical approximation results

Abstract

Motivated by classical results of approximation theory, we define an Hermite-type interpolation in terms of $n$-dimensional subspaces of the space of $n$ times continuously differentiable functions. In the main result of this paper, we establish an error term in integral form for this interpolation in the case when the $n$-dimensional subspace is the kernel of an $n$th order linear differential operator with constant coefficients. Several corollaries are deduced illustrating the applicability of this result.