On dimension elevation in Quasi Extended Chebyshev spaces

Via blossoms we analyse the dimension elevation process from to , where is spanned over [0, 1] by 1, x,..., x ^n-2, x ^p , (1 − x) ^q , p, q being any convenient real numbers. Such spaces are not Extended Chebyshev spaces but Quasi Extended Chebyshev spaces. They were recently introduced in CAGD for shape preservation purposes (Costantini in Math Comp 46:203–214; 1986, Costantini in Advanced Course on FAIRSHAPE, pp. 87–114 in 1996; Costantini in Curves and Surfaces with Applications in CAGD, pp. 85–94, 1997). Our results give a new insight into the special case p = q for which dimension elevation had already been considered, first when p = q was supposed to be an integer (Goodman and Mazure in J Approx Theory 109:48–81, 2001), then without the latter requirement (Costantini et al. in Numer Math 101:333–354, 2005). The question of dimension elevation in more general Quasi Extended Chebyshev spaces is also addressed.     

Extended Chebyshev spaces in rationality

On a closed bounded interval, a given Extended Chebyshev space can be defined by means of generalised derivatives associated with systems of weight functions. Only recently we could identify all such systems, describing an iterative process to build them. In the present work, we interpret the first step of this process as the construction of rational spaces based on Extended Chebyshev spaces. This construction establishes an interesting symmetry between all Extended Chebyshev spaces “good for design” (i.e., all those which contain constants and which possess blossoms) and the rational spaces based on them (Extended Chebyshev spaces in rationality). In particular, this symmetry results in a very simple relation between the corresponding blossoms. A special case is obtained when considering polynomial spaces as examples of Extended Chebyshev spaces. The classical rational spaces then appear as examples of Extended Chebyshev spaces good for design, that is, possessing blossoms. This offers interesting new insights on the famous so-called rational Bézier curves.     

Various characterisations of Extended Chebyshev spaces via blossoms

Among all W-spaces (i.e. spaces with nonvanishing Wronskians), extended Cheyshev spaces can be characterised by the existence of either Bernstein bases, or B-spline bases, or Bézier points, or blossoms in the spaces obtained by integration. To cite this article: M.-L. Mazure, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

We study the existence and shape preserving properties of a generalized Bernstein operator B _n fixing a strictly positive function f ₀, and a second function f ₁ such that f ₁/f ₀ is strictly increasing, within the framework of extended Chebyshev spaces U _n . The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B _n : C[a, b] → U _n with strictly increasing nodes, fixing f₀, f₁ ? U_n{f_{0}, f_{1} \in U_{n}} . If U_n ì U_{n + 1}{U_{n} \subset U_{n + 1}} and U _n+1 has a non-negative Bernstein basis, then there exists a Bernstein operator B _n+1 : C[a, b] → U _n+1 with strictly increasing nodes, fixing f ₀ and f ₁. In particular, if f ₀, f ₁, . . . , f _n is a basis of U _n such that the linear span of f ₀, . . . , f _k is an extended Chebyshev space over [a, b] for each k = 0, . . . , n, then there exists a Bernstein operator B _n with increasing nodes fixing f ₀ and f ₁. The second main result says that under the above assumptions the following inequalities hold

拟第一可数空间和弱拟第一可数空间

讨论了拟第一可数空间和弱拟第一可数空间的遗传性和可积性,给出了一些反例来说明这两类空间在某些拓扑运算下的不封闭性,同时研究了它们具有遗传性和可积性的充要条件.     

Some new Chebyshev spaces

In this note we see another circumstance where Chebyshev polynomials play a significant role. In particular, we present some new extended Chebyshev spaces that arise in the asymptotic stability of the zero solution of first order linear delay differential equations with m commensurate delays where a_j,j=0,…,m, are constants and τ>0 is constant.     

Interpolatory quadrature formulas on the unit circle for Chebyshev weight functions

Summary. In this paper, interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions. Positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands. A comparison is made with Szeg? quadrature formulas. Finally, an application to the interval [-1,1] is also carried out. Received February 29, 2000 / Published online August 17, 2001     

On Chebyshev functions and Klee functions

Associated to a lower semicontinuous function, one can define its proximal mapping and farthest mapping. The function is called Chebyshev (Klee) if its proximal mapping (farthest mapping) is single-valued everywhere. We show that the function f is 1/λ-hypoconvex if its proximal mapping Pλf is single-valued. When the function f is bounded below, and Pλf is single-valued for every λ>0, the function must be convex. Similarly, we show that the function f is 1/μ-strongly convex if the farthest mapping Qμf is single-valued. When the function is the indicator function of a set, this recovers the well-known Chebyshev problem and Klee problem in Rn. We also give an example illustrating that a continuous proximal mapping (farthest mapping) needs not be locally Lipschitz, which answers one open question by Hare and Poliquin.     

The error norm of Gauss-Radau quadrature formulae for Chebyshev weight functions

In certain spaces of analytic functions the error term of the Gauss-Radau quadrature formula relative to a (nonnegative) weight function is a continuous linear functional. We compute or estimate the norm of the error functional for any one of the four Chebyshev weight functions.     

Bernstein-type operators in Chebyshev spaces

We prove that it is possible to construct Bernstein-type operators in any given Extended Chebyshev space and we show how they are connected with blossoms. This generalises and explains a recent result by Aldas/Kounchev/Render on exponential spaces. We also indicate why such operators automatically possess interesting shape preserving properties and why similar operators exist in still more general frameworks, e.g., in Extended Chebyshev Piecewise spaces. We address the problem of convergence of infinite sequences of such operators, and we do prove convergence for special instances of Müntz spaces.      

Chebyshev spaces with polynomial blossoms

The use of extended Chebyshev spaces in geometric design is motivated by the interesting shape parameters they provide. Unfortunately the algorithms such spaces lead to are generally complicated because the blossoms themselves are complicated. In order to make up for this inconvenience, we here investigate particular extended Chebyshev spaces, containing the constants and power functions whose exponents are consecutive positive integers. We show that these spaces lead to simple algorithms due to the fact that the blossoms are polynomial functions. Furthermore, we also describe an elegant dimension elevation algorithm which makes it possible to return to polynomial spaces and therefore to use all the classical algorithms for polynomials. This revised version was published online in June 2006 with corrections to the Cover Date.     

Chebyshev inequalities in symmetric spaces

The characterization (by means of inequalities) of some special Banach spaces is investigated.Translated from Matematicheskie Zametki, Vol. 10, No. 2, pp. 195–205, August, 1971.The authors wish to thank the reviewer for his interest in this work and for his valuable advice.     

Finding all systems of weight functions associated with a given extended Chebyshev space

Systems of weight functions and corresponding generalised derivatives are classically used to build extended Chebyshev spaces on a given interval. This is a well-known procedure. Conversely, if the interval is closed and bounded, it is known that a given extended Chebyshev space can always be associated with a system of weight functions via the latter procedure. In the present article we determine all such possibilities, that is, all systems of weight functions which can be used to define a given extended Chebyshev space on a closed bounded interval.     

Estimates for then-widths of compact sets of differentiate functions in spaces with weight functions

We obtain two-sided estimates of the widths defined by A. N. Kolmogorov for the unit spheres of the anisotropic Sobolev-Slobodetskii spaces in L_q() for an arbitrary measure, guaranteeing the compactness of the corresponding embedding. As shown by examples, these estimates turn out to be exact as far as the order is concerned for measures with a strong singularity and, in addition, they allow us to justify the formula of the classical spectral asymptotics under very weak (close to necessary) restrictions of the measure .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 59, pp. 117–132, 1976.