  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.
