The paper studies polynomial approximation models with a new type of constraints that enable to get estimates with significant properties. Recently we enhanced a representation of polynomials based on three reference points. Here we propose a two-part cubic smoothing scheme that leverages this representation. The presence of these points in the model has several consequences. The most important one is the fact that by appropriate location of the reference points the resulting approximant of two successively assessed neighboring approximants will be smooth. We also show that the considered models provide estimates with appropriate statistical properties such as consistency and asymptotic normality.
Interpolating and approximating polynomials have been living separately more than two centuries. Our aim is to propose a general parametric regression model that incorporates both interpolation and approximation. The paper introduces first a new r-point transformation that yields a function with a simpler geometrical structure than the original function. It uses r≥2 reference points and decreases the polynomial degree by r−1. Then a general representation of polynomials is proposed based on r≥1 reference points. The two-part model, which is suited to piecewise approximation, consist of an ordinary least squares polynomial regression and a reparameterized one. The later is the central component where the key role is played by the reference points. It is constructed based on the proposed representation of polynomials that is derived using the r-point transformation Tr(x). The resulting polynomial passes through r reference points and the other points approximates. Appropriately chosen reference points ensure quasi smooth transition between the two components and decrease the dimension of the LS normal matrix. We show that the model provides estimates with such statistical properties as consistency and asymptotic normality.