Rachid Ait-Haddou

Chebyshev blossoming in Müntz spaces: Toward shaping with Young diagrams

The notion of a blossom in extended Chebyshev spaces offers adequate generalizations and extra-utilities to the tools for free-form design schemes. Unfortunately, such advantages are often overshadowed by the complexity of the resulting algorithms. In this work, we show that for the case of Muntz spaces with integer exponents, the notion of a Chebyshev blossom leads to elegant algorithms whose complexities are embedded in the combinatorics of Schur functions. We express the blossom and the pseudo-affinity property in Muntz spaces in terms of Schur functions. We derive an explicit expression for the Chebyshev-Bernstein basis via an inductive argument on nested Muntz spaces. We also reveal a simple algorithm for dimension elevation. Free-form design schemes in Muntz spaces with Young diagrams as shape parameters are discussed.

Explicit Gaussian quadrature rules for C 1 cubic splines with symmetrically stretched knot sequences

We provide explicit expressions for quadrature rules on the space of C 1 cubic splines with non-uniform, symmetrically stretched knot sequences. The quadrature nodes and weights are derived via an explicit recursion that avoids an intervention of any numerical solver and the rule is optimal, that is, it requires minimal number of nodes. Numerical experiments validating the theoretical results and the error estimates of the quadrature rules are also presented. The paper provides expressions for quadrature rules on the space of C 1 cubic splines with non-uniform, symmetrically stretched knot sequences.Any function from the space is exactly integrated by the rule.The quadrature nodes and weights are derived via explicit recursion formulae, with no intervention of any numerical solver.The rule is optimal, i.e. it requires minimal number of nodes.

Dimension elevation in Müntz spaces: A new emergence of the Müntz condition

Abstract We show that the limiting polygon generated by the dimension elevation algorithm with respect to the Muntz space s p a n ( 1 , t r 1 , t r 2 , … , t r m , … ) , with 0 r 1 r 2 ⋯ r m ⋯ and lim n → ∞ r n = ∞ , over an interval [ a , b ] ⊂ ] 0 , ∞ [ converges to the underlying Chebyshev–Bezier curve if and only if the Muntz condition ∑ i = 1 ∞ 1 r i = ∞ is satisfied. The surprising emergence of the Muntz condition in the problem raises the question of a possible connection between the density questions of nested Chebyshev spaces and the convergence of the corresponding dimension elevation algorithms. The question of convergence with no condition of monotonicity or positivity on the pairwise distinct real numbers r i remains an open problem.

Constrained multi-degree reduction with respect to Jacobi norms

We show that a weighted least squares approximation of Bezier coefficients with factored Hahn weights provides the best constrained polynomial degree reduction with respect to the Jacobi L 2 -norm. This result affords generalizations to many previous findings in the field of polynomial degree reduction. A solution method to the constrained multi-degree reduction with respect to the Jacobi L 2 -norm is presented. We study the problem of constrained degree reduction with respect to Jacobi norms.Our results generalize several previous findings on polynomial degree reduction.We explore the space of Jacobi parameters on the reduced polynomial approximation.

Pythagorean hodograph spline spirals that match G 3 Hermite data from circles

A construction is given for a G 3 piecewise rational Pythagorean hodograph convex spiral which interpolates two G 3 Hermite data associated with two non-concentric circles, one being inside the other. The spiral solution is of degree 7 and is the involute of a G 2 convex curve, referred to as the evolute solution, with prescribed length, and composed of two PH quartic curves. Conditions for G 3 continuous contact with circles are then studied and it turns out that an ordinary cusp at each end of the evolute solution is required. Thus, geometric properties of a family of PH polynomial quartics, allowing to generate such an ordinary cusp at one end, are studied. Finally, a constructive algorithm is described with illustrative examples.

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

We study the best polynomial degree reduction with respect to the q-L2-norm.We study a finite analogue with respect to finite q-lattices.We present applications to q-orthogonal polynomials. We show that a weighted least squares approximation of q-Bezier coefficients provides the best polynomial degree reduction in the q-L2-norm. We also provide a finite analogue of this result with respect to finite q-lattices and we present applications of these results to q-orthogonal polynomials.

Continuous and discrete best polynomial degree reduction with Jacobi and Hahn weights

We show that the weighted least squares approximation of Bezier coefficients with Hahn weights provides the best polynomial degree reduction in the Jacobi L 2 -norm. A discrete analogue of this result is also provided. Applications to Jacobi and Hahn orthogonal polynomials are presented.

The Fundamental Blossoming Inequality in Chebyshev Spaces—I: Applications to Schur Functions

A classical theorem by Chebyshev says how to obtain the minimum and maximum values of a symmetric multiaffine function of n variables with a prescribed sum. We show that, given two functions in an Extended Chebyshev space good for design, a similar result can be stated for the minimum and maximum values of the blossom of the first function with a prescribed value for the blossom of the second one. We give a simple geometric condition on the control polygon of the planar parametric curve defined by the pair of functions ensuring the uniqueness of the solution to the corresponding optimization problem. This provides us with a fundamental blossoming inequality associated with each Extended Chebyshev space good for design. This inequality proves to be a very powerful tool to derive many classical or new interesting inequalities. For instance, applied to Müntz spaces and to rational Müntz spaces, it provides us with new inequalities involving Schur functions which generalize the classical MacLaurin’s and Newton’s inequalities. This work definitely demonstrates that, via blossoms, CAGD techniques can have important implications in other mathematical domains, e.g., combinatorics.

On the Lorentz degree of a product of polynomials

In this note, we negatively answer two questions of T. Erdelyi (1991, 2010) on possible lower bounds on the Lorentz degree of product of two polynomials. We show that the correctness of one question for degree two polynomials is a direct consequence of a result of Barnard et al. (1991) on polynomials with nonnegative coefficients.