Improved Interpolation and Approximation through Order Manipulation

Abstract

A novel transform, which can improve interpolation and reduce approximation error, is introduced in this paper. This method can be applied to various formulas, including interpolation and approximation methods, which are denoted in the process of order manipulation. Subsequently, the paper shows how to achieve higher degree polynomial approximations through fewer interpolation points, which is impossible with ordinary methods of interpolation. In fact, this leads to an alternative solution to oscillatory behavior and Runge’s phenomenon, occurring in polynomial interpolation, or methods of approximation of least squares, when the number of points is increased significantly to achieve higher degree polynomials with the aim of error reduction. Several ideas—in the form of theorems and their proofs—are also studied on the basis of the smoothing process of the interpolation. Finally, a comprehensive comparison, with the aim of showing the advantage of the new methods in the form of MSE vs number of samples, is provided.