Vandermonde type determinants and blossoming

For extended Chebyshev spaces spanned by power functions, the blossoms can be expressed by means of Vandermonde type determinants. When the exponents are nonnegative integers, it is possible to use the classical algorithms for polynomial functions after one or several dimension elevation processes. This provides interesting shape parameters. This revised version was published online in June 2006 with corrections to the Cover Date.     

Four Properties to Characterize Chebyshev Blossoms

The blossom of a polynomial function of degree less than or equal to n is known as the unique function of n variables to be symmetric, affine with respect to each variable, and to coincide with the polynomial function itself when all the variables are equal. Chebyshev blossoms do satisfy similar properties, the affinity being now replaced by a pseudoaffinity property with respect to each variable. However, by themselves, these three properties may be insufficient to clearly identify the blossom of a given function. In this paper we show that this identification is made possible through an additional appropriate requirement of differentiability. June 15, 1999. Date revised: January 20, 2000. Date accepted: May 8, 2000.     

Pseudoaffinity, de Boor algorithm, and blossoms

In order to ensure existence of a de Boor algorithm (hence of a B-spline basis) in a given spline space with (n+1)-dimensional sections, it is important to be able to generate each spline by restriction to the diagonal of a symmetric function of n variables supposed to be pseudoaffine w.r. to each variable. We proved that a way to obtain these three properties (symmetry, n-pseudoaffinity, diagonal property) is to suppose the existence of blossoms on the set of admissible n-tuples, given that blossoms are defined in a geometric way by means of intersections of osculating flats. In the present paper, we examine the converse: do symmetry, n-pseudoaffinity, and diagonal property imply existence of blossoms?     

Chebyshev splines beyond total positivity

For polynomial splines as well as for Chebyshev splines, it is known that total positivity of the connection matrices is sufficient to obtain B-spline bases. In this paper we give a necessary and sufficient condition for the existence of B-bases (or, equivalently, of blossoms) for splines with connection matrices and with sections in different four-dimensional extended Chebyshev spaces.     

Bernstein bases in Müntz spaces

Extended Chebyshev spaces possess Bernstein type bases. In this paper, we determine the expressions of such bases in spaces spanned by the constants and power functions with consecutive integer exponents. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

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. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Chebyshev polynomial method for line integrals with singularities

In this paper we consider a Chebyshev polynomial method for the calculation of line integrals along curves with Cauchy principal value or Hadamard finite part singularities. The major point we address is how to reconstruct the value of the integral when the parametrization of the curve is unknown and only empirical data are available at some discrete set of nodes. We replace the curve by a near‐minimax parametric polynomial approximation, and express the integrand by means of a sum of Chebyshev polynomials. We make use of a mapping property of the Hadamard finite part operator to calculate the value of the integral. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.      

Blossoming: A Geometrical Approach

A geometrical approach of a notion of blossom for piecewise smooth Chebyshev functions is developed by considering convenient intersections of osculating flats. A subblossoming principle allows us to obtain all the expected properties and leads to the notion of blossom for splines based on a given piecewise smooth Chebyshev function. January 7, 1997. Date revised: October 1, 1997. Date accepted: December 22, 1997.     

Numerical solution of the Poisson equation over hypercubes using reduced Chebyshev polynomial bases

We describe how to use new reduced size polynomial approximations for the numerical solution of the Poisson equation over hypercubes. Our method is based on a non-standard Galerkin method which allows test functions which do not verify the boundary conditions. Numerical examples are given in dimensions up to 8 on solutions with different smoothness using the same approximation basis for both situations. A special attention is paid on conditioning problems.     

Multivariate Markov polynomial inequalities and Chebyshev nodes

This article considers the extension of V.A. Markov's theorem for polynomial derivatives to polynomials with unit bound on the closed unit ball of any real normed linear space. We show that this extension is equivalent to an inequality for certain directional derivatives of polynomials in two variables that have unit bound on the Chebyshev nodes. We obtain a sharpening of the Markov inequality for polynomials whose values at specific points have absolute value less than one. We also obtain an interpolation formula for polynomials in two variables where the interpolation points are Chebyshev nodes.     

Chebyshev Spaces and Bernstein Bases

We characterize extended Chebyshev spaces by the fact that any Hermite interpolation problem involving at most two different points has a unique solution. This enables us to prove that, in a given space, Bernstein bases exist if and only if the space obtained by differentiation is an extended Chebyshev space.     

Blossoming stories

We review the main properties of blossoms along with their important repercussions in all aspects of geometric design. Not only are they an elegant and efficient tool to express all classical algorithms, but they are also a fundamental concept, as proven by the fact that their existence is equivalent to the existence of B-spline bases. AMS subject classification 65D17     

Best quadrature formula on Sobolev class with Chebyshev weight

Using best interpolation function based on a given function information, we present a best quadrature rule of function on Sobolev class KW^r[-1,1]${\mathit{KW}}^r[-1,1]$ with Chebyshev weight. The given function information means that the values of a function f∈KW^r[-1,1]$f\in {\mathit{KW}}^r[-1,1]$ and its derivatives up to r-1$r-1$ order at a set of nodes x$\mathbf{x}$ are given. Error bounds are obtained, and the method is illustrated by some examples.     

Towards existence of piecewise Chebyshevian B-spline bases

The paper addresses the problem of how to ensure existence of blossoms in the context of piecewise spaces built from joining different extended Chebyshev spaces by means of connection matrices. The interest of this issue lies in the fact that existence of blossoms is equivalent to existence of B-spline bases in all associated spline spaces. As is now classical, blossoms are defined in a geometric way by means of intersections of osculating flats. In such a piecewise context, intersecting a number of osculating flats is a tough proposition. In the present paper, we show that blossoms exist if an only if Bézier points exist, which significantly simplifies the problem. Existence of blossoms also proves to be equivalent to existence of Bernstein bases. In order to establish the latter results, we start by extending to the piecewise context some results which are classical for extended Chebyshev spaces. AMS subject classification 65D17, 65D07     

Chebyshev polynomial approximation for high‐order partial differential equations with complicated conditions

In this article, a new method is presented for the solution of high‐order linear partial differential equations (PDEs) with variable coefficients under the most general conditions. The method is based on the approximation by the truncated double Chebyshev series. PDE and conditions are transformed into the matrix equations, which corresponds to a system of linear algebraic equations with the unknown Chebyshev coefficients, via Chebyshev collocation points. Combining these matrix equations and then solving the system yields the Chebyshev coefficients of the solution function. Some numerical results are included to demonstrate the validity and applicability of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Identities for piecewise polynomial spaces determined by connection matrices

Summary TwoB-spline results — Marsden's identity and the de Boor-Fix dual functionals — are extended to geometrically continuous curves determined by connection matrices.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Orthonormality of Cardinal Chebyshev B-Spline Bases in Weighted Sobolev Spaces

Extending a recent result of Ulrich Reif on cardinal polynomial B-splines, we show that the cardinal Chebyshev B-spline basis associated with a linear differential operator with constant real coefficients is orthonormal with respect to a unique weighted Sobolev-type inner product.     

On the stability of polynomial transformations between Taylor,Bernstein and Hermite forms

The stability of transformations between Taylor and Hermite and Bernstein and Hermite forms of the polynomials are investigated. The results are analogous to Farouki's concerning the stability of the transformation between Taylor and Bernstein form. An exact asymptotic is given for the condition numbers in thel ¹ case.Research was partially supported by the Copernicus grant RECCAD 94-1068 and by the National Research Foundation of the Hungarian Academy of Sciences grant 16420.