Local Polynomial Operator Interpolation for Tunable Rational Dynamical Models

Abstract

This paper introduces a local polynomial operator interpolation framework for constructing tunable rational dynamical models between selected integer-order regimes. For a fixed interpolation degree , the operator  is defined as a finite linear combination of classical integer-order derivatives with Lagrange coefficients depending on a real parameter . The construction recovers the classical derivatives at the interpolation nodes, remains finite-dimensional, and satisfies basic stability estimates on fixed stencils. Since its Laplace symbol  is a polynomial in , equations involving  reduce to finite-order ordinary differential equations and yield rational transfer functions. The proposed operator is a local rational alternative for applications where tunability, interpretability, and computational simplicity are more important than long-memory modelling.