Interpolation at a Few Points

Wulbert, Daniel

doi:10.1006/jath.1998.3224

Cited by 6 publications

(5 citation statements)

References 7 publications

(1 reference statement)

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“…The existence of k-regular maps is also important in approximation theory, by their connection with interpolation spaces [Han80,Wul99,She04,She09]. Let V be a subspace of continuous functions on R m , then we say that V is a k-interpolation space if for every set Z of k distinct points on R m and every function f : Z → R, there is a function g ∈ V such that f (z) = g(z) for every z ∈ Z.…”

Section: Introductionmentioning

confidence: 99%

Constructions of $k$-regular maps using finite local schemes

et al. 2019

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A continuous map R m → R N or C m → C N is called k-regular if the images of any k points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk is to determine the minimal value of N for which such maps exist. The methods of algebraic topology provide lower bounds for N , however there are very few results on the existence of such maps for particular values m and k. Using the methods of algebraic geometry we construct k-regular maps. We relate the upper bounds on N with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for k ≤ 9, and we provide explicit examples for k ≤ 5. We also provide upper bounds for arbitrary m and k.

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Section: Introductionmentioning

confidence: 99%

Constructions of $k$-regular maps using finite local schemes

et al. 2019

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“…Some lower bounds for Lagrange interpolation are given in [3,12,14,15]. Yet, the exact values of the minimal dimension of a space that interpolates at five points in R 2 or four points in R 3 are not known.…”

Section: Discussionmentioning

confidence: 99%

“…For Lagrange interpolation several results of this type are known (cf. [3,[12][13][14][15]). As far as I know, this is the first such study for Hermite (Lagrange) Interpolation.…”

Section: Introduction and { U V }-Interpolating Casementioning

confidence: 99%

Case study in bivariate Hermite interpolation

Shekhtman

2005

Journal of Approximation Theory

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In this article we investigate the minimal dimension of a subspace of C 1 (R 2 ) needed to interpolate an arbitrary function and some of its prescribed partial derivatives at two arbitrary points. The subspace in question may depend on the derivatives, but not on the location of the points. Several results of this type are known for Lagrange interpolation. As far as I know, this is the first such study for Hermite Interpolation.

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“…An Euclidean-continuous map f : R n → R N or f : C n → C N is called r-regular if the images of every r points are linearly independent. The existence of r-regular maps to C N for given (r, n) is highly nontrivial and has attracted the interest of algebraic topologists, including Borsuk [Haa17, Kol48, Bor57, Chi79, CH78, Han96, HS80, Vas92], and interpolation theorists [Han80,Wul99,She04,She09]. Their developments improved the lower bounds on N , depending on n and r, see [BCLZ16], but few examples or sharp upper bounds were known.…”

Section: Application To Constructing R-regular Mapsmentioning

confidence: 99%

Hilbert schemes of points and applications

Jelisiejew¹

2022

Preprint

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Long, long ago 1 , in a galaxy 2 far, far away 3 ...The following is the author's PhD Thesis, defended in September 2017. It has not been edited since (except for the footnotes in the open problems section and for this note). In particular, there is nothing here about applications of Białynicki-Birula decomposition to Hilbert schemes [Jel19, Jel20, SS21], the multigraded Hilbert schemes and border apolarity [BB21, Mań20], fiber-full Hilbert schemes [CRR21, CRR22] and other recent developments. Philosophically speaking, is it reassuring that there are too many new interesting papers on Hilbert schemes of points to summarize here. It is also very interesting that many of the open problems listed below in §1.5 remain actual. Today, the author would add also the more applied open problems (see [JLP22]): a large part of the classical problem of determining the complexity of matrix multiplication can be reduced fruitfully to the study of the Hilbert scheme of points.While the author is not aware of any errors, it is likely they exist here. This work is put on arXiv due/thanks to several people who decided to refer to it and only to grant this piece a quasi-permanent place to rest.1 That is, five years 2 Metaphorically speaking 3 In terms of scientific development ¨

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Interpolation at a Few Points

Cited by 6 publications

References 7 publications

Constructions of $k$-regular maps using finite local schemes

Constructions of $k$-regular maps using finite local schemes

Case study in bivariate Hermite interpolation

Hilbert schemes of points and applications

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