# Simultaneous Approximation of Measurement Values and Derivative Data using Discrete Orthogonal Polynomials

## TL;DR

This paper introduces a stable, efficient polynomial approximation method that simultaneously fits measurement values and derivatives using a weighted Hermite approach with orthogonal polynomials, suitable for real-time applications.

## Contribution

The paper develops a numerically stable, recurrence-based Hermite approximation method that integrates value and derivative data with covariance weighting, improving efficiency and stability over existing techniques.

## Key findings

- The method achieves higher numerical stability in polynomial fitting.
- It allows real-time approximation using inner products instead of matrix inverses.
- Monte Carlo simulations validate the approach on synthetic data.

## Abstract

This paper presents a novel method for polynomial approximation (Hermite approximation) using the fusion of value and derivative information. Therefore, the least-squares error in both domains is simultaneously minimized. A covariance weighting is used to introduce a metric between the value and derivative domain, to handle different noise behaviour. Based on a recurrence relation with full re-orthogonalization, a weighted polynomial basis function set is generated. This basis is numerically more stable compared to other algorithms, making it suitable for the approximation of data with high degree polynomials. With the new method, the fitting problem can be solved using inner products instead of matrix-inverses, yielding a computational more efficient method, e.g., for real-time approximation. A Monte Carlo simulation is performed on synthetic data, demonstrating the validity of the method. Additionally, various tests on the basis function set are presented, showing the improvement on the numerical stability.