A system of differential equations connected with a polynomial class of shifts of a box spline

Akopyan, A. A.; Saakyan, A. A.

doi:10.1007/bf01158024

Cited by 15 publications

(30 citation statements)

References 6 publications

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“…It reflects in its structure much of the geometry of the multiset X, and was independently discovered by several authors ([HS1,J,DR]; some of us regret that this is the chronological order; see also [DM]). Now note that l L is a homogeneous polynomial of (exact) degree *L, and is constant in all directions perpendicular to ran…”

Section: The Space P S (X )mentioning

confidence: 95%

Polynomial Interpolation to Data on Flats in Rd

Boor

Dyn

Ron

2000

Journal of Approximation Theory

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Stimulated by recent work of Hakopian and Sahakian, polynomial interpolation to data at all the s-dimensional intersections of an arbitrary sequence of hyperplanes in R d is considered, and reduced, by the adjunction of an additional s hyperplanes in general position with respect to the given sequence, to the case s=0 solved much earlier by two of the present authors. In particular, interpolation is from the very same polynomial spaces already used earlier. The difficult question of multiplicity and corresponding matching of derivative information is completely solved, with the number of independent derivative conditions at an intersection exactly equal to that intersection's multiplicity. Also, the consistency requirements placed on the data are minimal in the sense that they need to be checked only at the finitely many 0-dimensional intersections of the hyperplanes involved. The arguments used provide, incidentally, further insights into the two polynomial spaces, P(5) and D(5), of basic interest in box spline theory. Academic Press

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Section: The Space P S (X )mentioning

confidence: 95%

Polynomial Interpolation to Data on Flats in Rd

Boor

Dyn

Ron

2000

Journal of Approximation Theory

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“…Here, [1] means that the degree of the graded vector space should be shifted up by one. Theorem 2.15 (Theorems 11.35 and 11.37 in [21]).…”

Section: Recall That Z[λ] Acts Via Translation Onmentioning

confidence: 99%

Splines, lattice points, and arithmetic matroids

Lenz

2015

J Algebr Comb

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Abstract. Let X be a (d × N )-matrix. We consider the variable polytope Π X (u) = {w ≥ 0 : Xw = u}. It is known that the function T X that assigns to a parameter u ∈ R d the volume of the polytope Π X (u) is piecewise polynomial. The Brion-Vergne formula implies that the number of lattice points in Π X (u) can be obtained by applying a certain differential operator to the function T X . In this article we slightly improve the Brion-Vergne formula and we study two spaces of differential operators that arise in this context: the space of relevant differential operators (i. e. operators that do not annihilate T X ) and the space of nice differential operators (i. e. operators that leave T X continuous). These two spaces are finite-dimensional homogeneous vector spaces and their Hilbert series are evaluations of the Tutte polynomial of the arithmetic matroid defined by the matrix X. They are closely related to the P-spaces studied by ArdilaPostnikov and Holtz-Ron in the context of zonotopal algebra and power ideals.

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“…The space P(X) first appeared in approximation theory [1,10,16]. Later, spaces of this type and generalisations were also studied by authors in other fields, e. g. [2,4,19,20,21,24].…”

Section: Introductionmentioning

confidence: 99%

Interpolation, Box Splines, and Lattice Points in Zonotopes

Lenz¹

2013

International Mathematics Research Notices

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Abstract. Let X be a totally unimodular list of vectors in some lattice. Let B X be the box spline defined by X. Its support is the zonotope Z(X). We show that any real-valued function defined on the set of lattice points in the interior of Z(X) can be extended to a function on Z(X) of the form p(D)B X in a unique way, where p(D) is a differential operator that is contained in the so-called internal P-space. This was conjectured by Olga Holtz and Amos Ron. We also point out connections between this interpolation problem and matroid theory, including a deletion-contraction decomposition.

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A system of differential equations connected with a polynomial class of shifts of a box spline

Cited by 15 publications

References 6 publications

Polynomial Interpolation to Data on Flats in Rd

Polynomial Interpolation to Data on Flats in Rd

Splines, lattice points, and arithmetic matroids

Interpolation, Box Splines, and Lattice Points in Zonotopes

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