Pierre-Jean Laurent

On simultaneous approximation

In this exposition, we investigate in an extended framework the problem of simultaneous best approximation. Refinement of a formula for the subdifferential of restricted radius of a p-bounded set leads us to explore some interesting results on strong uniqueness of simultaneous best approximants. In particular, we also. deal here with simultaneous best approximation of certain sets of vector-valued functions and establish strong uniqueness of order 2 for such approximants under suitable Haar-like hypotheses.

Polynomial Chebyshev splines

A particular type of Chebyshev spaces included in polynomial spaces is studied leading to new shape parameters. Chebyshev splines based on these spaces are defined and the corresponding shape effects are analyzed.

Non-affine blossoms and subdivision for Q -splines

Abstract A parametric Q-spline curve is a C2 piecewise polynomial function of degree 4 for which each polynomial section satisfies an affine relation. For such a curve, a non-affine blossom can be defined and a corresponding non-affine subdivision algorithm can be derived.

Chebyshev splines and shape parameters

A parametric spline curve is defined whose restriction to each sub-interval belongs to a 4-dimensional piecewise Chebyshev subspace depending on coefficients which play the role of shape parameters.

A characterization by optimization of the orthocenter of a triangle

“Nothing in the world takes place without optimization, and there is no doubt that all aspects of the world that have a rational basis can be explained by optimization methods” used to say L.Euler in 1744. This is confirmed by numerous applications of mathematics in physics, mechanics, economy, etc. In this note, we show that it is also the case for the classical centers of the triangle, more specifically for the orthocenter. To our best knowledge, the characterization of the orthocenter of a triangle by optimization, that we are going to present, is new. Let us begin by revisiting the usual centers of a triangle and their characterizations by optimization. Let T = ABC be a triangle (the points A,B,C are supposed non aligned, of course). The centroid or isobarycenter of T is the point of concurrence of the medians of T , which are the line-segments joining the vertices of T to the mid-points of the respective opposite sides. It is the point which minimizes on T the following objective function or criterion P 7−→ (PA) + (PB) + (PC). (1)

Pseudo-cubic weighted splines can be C 2 or G 2

Publisher Summary This chapter discusses the minimization of pseudo cubic weighted splines that is continuous and piecewise linear on the subdivision. The solution is a C 2 quartic spline, but surprisingly it has, in fact, all the properties of the classical cubic spline: computing the solution leads to a symmetric tridiagonal linear system, computing the corresponding smoothing spline leads to a block 2 × 2 tridiagonal system, and the associated B-spline is based on 4 intervals of the subdivision. This is because of the fact that, on each subinterval, the polynomial of degree 4 satisfies a linear relation depending on q i . This relation can be combined with G 2 continuity leading to a spline with 3 shape parameters at each knot.

A Characterization by Optimization of the Monge Point of a Tetrahedron

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Piecewise Smooth Spaces in Duality

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