Legendre–Bernstein basis transformations

The Bernstein form of a polynomial offers valuable insight into its geometrical behavior, and has thus won widespread acceptance as the basis for Bézier curves and surfaces. For least-squares approximation problems, on the other hand, the use of orthogonal bases, such as the Legendre polynomials, permits simple and efficient constructions for convergent sequences of approximants. We derive the transformation matrices that map the Bernstein and Legendre forms of a degree-n polynomial on [0,1] into each other, and examine the stability of this linear map. In the p=1 and ∞ norms, the condition number of the Legendre–Bernstein transformation matrix grows at a significantly slower rate with n than in the well-studied power-Bernstein case, and at a dramatically slower rate than for other common (e.g., Bernstein–Hermite or power-Hermite) basis conversions. The utility of Legendre representations in approximation problems, and their relatively stable transformation to Bernstein–Bézier form, argue for more widespread applications of Legendre methods in CAGD algorithms.