Rachid Ait-Haddou

Rachid Ait-Haddou, Yusuke Sakane, Taishin Nomura

The notion of 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 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 term of Schur functions. We derive an explicit expression of the Chebyshev-Bernstein basis via an inductive argument on nested Muntz spaces. We also reveal a simple algorithm for the dimension elevation process. Free-form design schemes in Muntz spaces with Young diagrams as shape parameter will be discussed.

Michael Bartoň, Rachid Ait-Haddou, Victor Manuel Calo

We provide explicit expressions for quadrature rules on the space of $C^1$ quintic splines with uniform knot sequences over finite domains. 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 the minimal number of nodes. For each of $n$ subintervals, generically, only two nodes are required which reduces the evaluation cost by $2/3$ when compared to the classical Gaussian quadrature for polynomials. Numerical experiments show fast convergence, as $n$ grows, to the "two-third" quadrature rule of Hughes et al. for infinite domains.

Rachid Ait-Haddou, Yusuke Sakane, Taishin Nomura

We show that the generalized Bernstein bases in Muntz spaces defined by Hirschman and Widder [7] and extended by Gelfond [6] can be obtained as limits of the Chebyshev-Bernstein bases in Muntz spaces with respect to an interval [a,1] as the real number, a, converges to zero. Such a realization allows for concepts of curve design such as de Casteljau algorithm, blossom, dimension elevation to be translated from the general theory of Chebyshev blossom in Muntz spaces to these generalized Bernstein bases that we termed here as Gelfond-Bernstein bases. The advantage of working with Gelfond-Bernstein bases lies in the simplicity of the obtained concepts and algorithms as compared to their Chebyshev-Bernstein bases counterparts.

Rachid Ait-Haddou, Yusuke Sakane, Taishin Nomura

By identifying a family of corner cutting schemes as a dimension elevation process of Gelfond-Bezier curves, we give a Muntz type condition for the convergence of the generated control polygons to the underlying curve. 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.

Rachid Ait-Haddou

The theory of polar forms of polynomials is used to provide for sharp bounds on the radius of the largest possible disc (absolute stability radius), and on the length of the largest possible real interval (parabolic stability radius), to be inscribed in the stability region of an explicit Runge-Kutta method. The bounds on the absolute stability radius are derived as a consequence of Walsh's coincidence theorem, while the bounds on the parabolic stability radius are achieved by using Lubinsky-Ziegler's inequality on the coefficients of polynomials expressed in the Bernstein bases and by appealing to a generalized variation diminishing property of Bezier curves. We also derive inequalities between the absolute stability radii of methods with different orders and number of stages.

Rachid Ait-Haddou, Michael Bartoň, Victor Manuel Calo

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.