Polynomial degree reduction in the discrete $$L_2$$ L 2 -norm equals best Euclidean approximation of h-Bézier coefficients

Abstract

We show that the best degree reduction of a given polynomial P from degree n to m with respect to the discrete L 2 -norm is equivalent to the best Euclidean distance of the vector of h-Bézier coefficients of P from the vector of degree raised h-Bézier coefficients of polynomials of degree m. Moreover, we demonstrate the adequacy of h-Bézier curves for approaching the problem of weighted discrete least squares approximation. Applications to discrete orthogonal polynomials are also presented.

Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials

“…Moreover, we show that just as in [2,3] with classical continuous and discrete orthogonal polynomials, combining the infinite and finite q-lattice degree reduction results described above provides a simple method for deriving formulae relating little q-Legendre polynomials and special cases of q-Hahn polynomials. Formulae expressing relations between little q-Jacobi polynomials and q-Hahn polynomials are also derived in [5,15] using a different approach.…”

“…This result was generalized by Ait-Haddou [2] by showing that the least squares approximation of h-Bézier coefficients provides the best polynomial degree reduction in the discrete L 2 -norm. Ait-Haddou also shows that combining the results on the continuous and the discrete degree reduction problems leads to a new and elegant method for deriving formulae relating Legendre orthogonal polynomials and discrete Legendre orthogonal polynomials [2]. Ait-Haddou [3] generalized this methodology by considering the problem of degree reduction with respect to both the Jacobi L 2 -norms and the discrete Hahn L 2 -norms.…”

“…The quality of the approximation depends on the norm. For several decades polynomial degree reduction has attracted considerable interest with respect to different norms: e.g., L ∞ -norm [8,24], L 2 -norm [14,18,19], L 1 -norm [12], L p -norm [6], Jacobi norms [2,25] as well as to different boundary constraints [1,16,17,21,26].…”

Constrained multi-degree reduction with respect to Jacobi norms

“…Optimal degree reduction is one of the fundamental tasks in Computer Aided Geometric Design (CAGD) and therefore has attracted researchers' attention for several decades [4,17,2,1,10,13,15]. Used not only for data compression, CAD/CAM software typically requires algorithms capable of converting a curve (surface) of a high degree to a curve (surface) of a lower degree.…”

Weighted G1-Multi-Degree Reduction of B´ezier Curves

“…al. in [1], and discrete cases have been studied in [2], [8]. The existing methods to find degree reduction have many issues including accumulate round-off errors, stability issues, complexity, accuracy, losing conjugacy, requiring the search direction to be set to the steepest descent direction frequently, experiencing ill-conditioned systems, leading to a singularity, and the most challenging difficulty is in applying the methods (difficulty and indirect).…”