Multivariate splines and polytopes

Xu, Zhiqiang

doi:10.1016/j.jat.2010.10.005

Cited by 9 publications

(5 citation statements)

References 19 publications

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“…Then, combining ( 23), ( 24) and ( 25), we obtain (20). It remains to prove (24) and (25). We first prove that…”

Section: The Asymptotic Property Of B-spline Frameletsmentioning

confidence: 90%

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On B-Spline Framelets Derived from the Unitary Extension Principle

Shen¹,

Xu²

2013

SIAM J. Math. Anal.

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The spline wavelet tight frames considered in [A. Ron and Z. Shen, J. Funct. Anal., 148 (1997), pp. 408-447] have been used widely in frame based image analysis and restorations (see, e.g., survey articles [B. Dong and Z. Shen, MRA-based wavelet frames and applications, IAS Lecture , there are few other properties of this family of tight frames that are currently known. The aim of this paper is to present a few new properties of this family that will provide some reasons why it is efficient in image analysis and restorations. In particular, we first present a recurrence formula for computing the generators of higher order spline wavelet tight frames from lower order ones. This simplifies the computations of the exact values of the functions in applications. Second, we represent each generator of spline wavelet tight frames as a certain order of derivative of some univariate box spline that satisfies a few additional properties. This is a crucial property used in [J.-F. Cai et al., J. Amer. Math. Soc., 25 (2012), pp. 1033-1089, where connections between total variational and wavelet frame based approaches for image restorations are established. Finally, we further show that each generator of sufficiently high order spline wavelet tight frames is close to a derivative of a properly scaled Gaussian function. This leads to the result that the wavelet system generated by finitely many consecutive derivatives of a properly scaled Gaussian function forms a frame whose frame bounds can be almost tight.

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“…Then, combining ( 23), ( 24) and ( 25), we obtain (20). It remains to prove (24) and (25). We first prove that…”

Section: The Asymptotic Property Of B-spline Frameletsmentioning

confidence: 90%

“…where D(R s ) is the test function space. The box spline can be consider as a volume function of the section of unit cubes (see [3,24,25]). If we take Ξ = (1, 1, .…”

Section: Representing ψ (M) ℓmentioning

confidence: 99%

On B-Spline Framelets Derived from the Unitary Extension Principle

Shen¹,

Xu²

2013

SIAM J. Math. Anal.

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“…The author would like to thank Olga Holtz and Martin Götze for discussions and helpful suggestions. He is grateful to Zhiqiang Xu for pointing out the paper [53].…”

Section: Organisation Of the Articlementioning

confidence: 99%

Zonotopal algebra and forward exchange matroids

Lenz

2016

Advances in Mathematics

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Zonotopal algebra is the study of a family of pairs of dual vector spaces of multivariate polynomials that can be associated with a list of vectors X. It connects objects from combinatorics, geometry, and approximation theory. The origin of zonotopal algebra is the pair (D(X), P(X)), where D(X) denotes the Dahmen-Micchelli space that is spanned by the local pieces of the box spline and P(X) is a space spanned by products of linear forms.The first main result of this paper is the construction of a canonical basis for D(X). We show that it is dual to the canonical basis for P(X) that is already known.The second main result of this paper is the construction of a new family of zonotopal spaces that is far more general than the ones that were recently studied by Ardila-Postnikov, Holtz-Ron, Holtz-Ron-Xu, Li-Ron, and others. We call the underlying combinatorial structure of those spaces forward exchange matroid. A forward exchange matroid is an ordered matroid together with a subset of its set of bases that satisfies a weak version of the basis exchange axiom.

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“…This implies, that vol(P A (b)) is a homogeneous polynomial of degree n in b. We refer to [13] for a closed formula of this polynomial. Martin Henk and Eva Linke, Institut für Algebra und Geometrie, Universität Magdeburg, Universitätsplatz 2, D-39106-Magdeburg, Germany E-mail address: martin.henk@ovgu.de, eva.linke@ovgu.de…”

Section: In General Pmentioning

confidence: 99%

Lattice points in vector-dilated polytopes

Henk,

Linke

2012

Preprint

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For A ∈ Z m×n we investigate the behaviour of the number of lattice points in PA(b) = {x ∈ R n : Ax ≤ b}, depending on the varying vector b. It is known that this number, restricted to a cone of constant combinatorial type of PA(b), is a quasi-polynomial function if b is an integral vector. We extend this result to rational vectors b and show that the coefficients themselves are piecewise-defined polynomials. To this end, we use a theorem of McMullen on lattice points in Minkowskisums of rational dilates of rational polytopes and take a closer look at the coefficients appearing there.

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Multivariate splines and polytopes

Cited by 9 publications

References 19 publications

On B-Spline Framelets Derived from the Unitary Extension Principle

On B-Spline Framelets Derived from the Unitary Extension Principle

Zonotopal algebra and forward exchange matroids

Lattice points in vector-dilated polytopes

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