Piecewise Extended Chebyshev Spaces: a numerical test for design

Given a number of Extended Chebyshev (EC) spaces on adjacent intervals, all of the same dimension, we join them via convenient connection matrices without increasing the dimension. The global space is called a Piecewise Extended Chebyshev (PEC) Space. In such a space one can count the total number of zeroes of any non-zero element, exactly as in each EC-section-space. When this number is bounded above in the global space the same way as in its section-spaces, we say that it is an Extended Chebyshev Piecewise (ECP) space. A thorough study of ECP-spaces has been developed in the last two decades in relation to blossoms, with a view to design. In particular, extending a classical procedure for EC-spaces, ECP-spaces were recently proved to all be obtained by means of piecewise generalised derivatives. This yields an interesting constructive characterisation of ECP-spaces. Unfortunately, except for low dimensions and for very few adjacent intervals, this characterisation proved to be rather difficult to handle in practice. To try to overcome this difficulty, in the present article we show how to reinterpret the constructive characterisation as a theoretical procedure to determine whether or not a given PEC-space is an ECP-space. This procedure is then translated into a numerical test, whose usefulness is illustrated by relevant examples

Interpolation of G1 Hermite data by C1 cubic-like sparse Pythagorean hodograph splines

open3siProvided that they are in appropriate configurations (tight data), given planar G1 Hermite data generate a unique cubic Pythagorean hodograph (PH) spline curve interpolant. On a given associated knot-vector, the corresponding spline function cannot be C1, save for exceptional cases. By contrast, we show that replacing cubic spaces by cubic-like sparse spaces makes it possible to produce infinitely many C1 PH spline functions interpolating any given tight G1 Hermite data. Such cubic-like sparse spaces involve the constants and monomials of consecutive degrees, and they have long been used for design purposes. Only lately they were investigated in view of producing PH curves and associated G1 PH spline interpolants with some flexibility. The present work strongly relies on these recent results.embargoed_20220415Ait-Haddou R.; Beccari C.V.; Mazure M.-L.Ait-Haddou R.; Beccari C.V.; Mazure M.-L

Bernstein operators for exponential polynomials

Let $L$ be a linear differential operator with constant coefficients of order $n$ and complex eigenvalues $\lambda_{0},...,\lambda_{n}$. Assume that the set $U_{n}$ of all solutions of the equation $Lf=0$ is closed under complex conjugation. If the length of the interval $[ a,b]$ is smaller than $\pi /M_{n}$, where M_{n}:=\max \left\{| \text{Im}% \lambda_{j}| :j=0,...,n\right\} , then there exists a basis $p_{n,k}$%, $k=0,...n$, of the space $U_{n}$ with the property that each $p_{n,k}$ has a zero of order $k$ at $a$ and a zero of order $n-k$ at $b,$ and each $$ is positive on the open interval $(a,b) .$ Under the additional assumption that $\lambda_{0}$ and $\lambda_{1}$ are real and distinct, our first main result states that there exist points $a=t_{0}<t_{1}<...<t_{n}=b$ and positive numbers $\alpha_{0},..,\alpha_{n}$%, such that the operator \begin{equation*} B_{n}f:=\sum_{k=0}^{n}\alpha_{k}f(t_{k}) p_{n,k}(x) \end{equation*} satisfies $B_{n}e^{\lambda_{j}x}=e^{\lambda_{j}x}$, for $j=0,1.$ The second main result gives a sufficient condition guaranteeing the uniform convergence of $B_{n}f$ to $f$ for each $f\in C[ a,b]$.Comment: A very similar version is to appear in Constructive Approximatio

Polynomial cubic splines with tension properties

In this paper we present a new class of spline functions with tension properties. These splines are composed by polynomial cubic pieces and therefore are conformal to the standard, NURBS based CAD/CAM systems

Matrix positivity preservers in fixed dimension. II: positive definiteness and strict monotonicity of Schur function ratios

We continue the study of real polynomials acting entrywise on matrices of fixed dimension to preserve positive semidefiniteness, together with the related analysis of order properties of Schur polynomials. Previous work has shown that, given a real polynomial with positive coefficients that is perturbed by adding a higher-degree monomial, there exists a negative lower bound for the coefficient of the perturbation which characterizes when the perturbed polynomial remains positivity preserving. We show here that, if the perturbation coefficient is strictly greater than this bound then the transformed matrix becomes positive definite given a simple genericity condition that can be readily verified. We identity a slightly stronger genericity condition that ensures positive definiteness occurs at the boundary. The analysis is complemented by computing the rank of the transformed matrix in terms of the location of the original matrix in a Schubert cell-type stratification that we have introduced and explored previously. The proofs require enhancing to strictness a Schur monotonicity result of Khare and Tao, to show that the ratio of Schur polynomials is strictly increasing along each coordinate on the positive orthant and non-decreasing on its closure whenever the defining tuples satisfy a coordinate-wise domination condition.Comment: 26 pages, no figures, LaTe