Bivariate ideal projectors and their perturbations

Shekhtman, Boris

doi:10.1007/s10444-007-9040-9

Cited by 7 publications

(3 citation statements)

References 12 publications

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“…Later, using linear algebra tools only, de Boor and Shekhtman [9] reproved the same result. Specifically, Shekhtman [10] completely analyzed the bivariate ideal projectors which are onto the space of polynomials of degree less than n over real or complex field, and verified the conjecture in this particular case.…”

Section: Introductionmentioning

confidence: 84%

Ideal Projectors of Type Partial Derivative and Their Perturbations

Li,

Zhang,

Dong

2011

Preprint

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In this paper, we verify Carl de Boor's conjecture on ideal projectors for real ideal projectors of type partial derivative by proving that there exists a positive η ∈ R such that a real ideal projector of type partial derivative P is the pointwise limit of a sequence of Lagrange projectors which are perturbed from P up to η in magnitude. Furthermore, we present an algorithm for computing the value of such η when the range of the Lagrange projectors is spanned by the Gröbner éscalier of their kernels w.r.t. lexicographic order.

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Section: Introductionmentioning

confidence: 84%

Ideal Projectors of Type Partial Derivative and Their Perturbations

Li,

Zhang,

Dong

2011

Preprint

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“…In addition, by Theorem 8 of [16], [15] also proved that certain low-rank multivariate ideal projectors are the limit of Lagrange projectors. Specifically, B. Shekhtman [17] completely analyzed the bivariate ideal projectors which are onto the space of polynomials of degree less than n over real or complex field, and verified the conjecture in this particular case.…”

Section: Introductionmentioning

confidence: 85%

On a C. de Boor's Conjecture in a Particular Case and Related Perturbation

Li¹,

Zhang²,

Dong³

2011

Preprint

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In this paper, we focus on two classes of D-invariant polynomial subspaces. The first is a classical type, while the second is a new class. With matrix computation, we prove that every ideal projector with each D-invariant subspace belonging to either the first class or the second is the pointwise limit of Lagrange projectors. This verifies a particular case of a C. de Boor's conjecture asserting that every complex ideal projector is the pointwise limit of Lagrange projectors. Specifically, we provide the concrete perturbation procedure for ideal projectors of this type.

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“…[3,22,23]). For a given bivariate ideal interpolant, suppose that ∆ is the finite set of interpolation condition functionals, M x , M y are its corresponding multiplication matrices.…”

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confidence: 97%

The discretization for bivariate ideal interpolation

Jiang

Zhang

Shang

2016

Journal of Computational and Applied Mathematics

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a b s t r a c tCarl de Boor conjectured that every ideal interpolant over complex field is the pointwise limit of Lagrange interpolants. Boris Shekhtman proved that the conjecture is true in two variables, and he also provided a counterexample for more than three variables. However, Shekhtman only mentioned the existence of some mathematical objects (without giving a method to compute them) in his proof of the bivariate case. For general interpolation condition functionals on the interpolation sites, we improve Shekhtman's method to find a sequence of interpolation sites (also called discrete sites), such that the corresponding Lagrange interpolants converge to the given bivariate ideal interpolant. We discuss a special case where the multiplicity space is of breadth one. The results in this paper give a completely algorithmic way to realize Shekhtman's method.

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Bivariate ideal projectors and their perturbations

Abstract: In this paper we present a complete description of ideal projectors from the space of bivariate polynomials F[x, y] onto its subspace F

Cited by 7 publications

References 12 publications

Ideal Projectors of Type Partial Derivative and Their Perturbations

Ideal Projectors of Type Partial Derivative and Their Perturbations

On a C. de Boor's Conjecture in a Particular Case and Related Perturbation

The discretization for bivariate ideal interpolation

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