Sur la représentation d'une population infinie par un nombre fini d'éléments

Tóth, László

doi:10.1007/bf02024494

Cited by 35 publications

(9 citation statements)

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“…In dimension d = 2, asymptotic results on the centroidal Voronoi tessellation figuring in Theorem 2.1 (or in the case p = 1 of Theorem 2.10) have also been obtained by various other authors; see, e.g., [5,6,11]. For large n, the Voronoi sets of the centroidal tesselation are approximately congruent regular hexagons.…”

Section: Corollary 29mentioning

confidence: 68%

Construction of positive definite cubature formulae and approximation of functions via Voronoi tessellations

Guessab

Schmeißer

2008

Adv Comput Math

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Let ⊂ R d be a compact convex set of positive measure. In a recent paper, we established a definiteness theory for cubature formulae of order two on . Here we study extremal properties of those positive definite formulae that can be generated by a centroidal Voronoi tessellation of . In this connection we come across a class of operators of the formwhere y 1 , . . . , y n are distinct points in and {φ 1 , . . . , φ n } is a partition of unity on . We present best possible pointwise error estimates and describe operators L n with a smallest constant in an L p error estimate for 1 ≤ p < ∞. For a generalization, we introduce a new type of Voronoi tessellation in terms of a twice continuously differentiable and strictly convex function f . It allows us to describe a best operator L n for approximating f by L n [ f ] with respect to the L p norm.Keywords Cubature formulae · Positive definite formulae · Multivariate approximation · Partitions of unity · Centroidal Voronoi tessellation · Generalized Voronoi tessellation Mathematics Subject Classifications (2000) Primary 05B45 · 41A63 · 41A80 · 52C22 · 65D30 · 65D32 · Secondary 06A06 · 26B25 · 26D15 · 52A40 Communicated by

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Section: Corollary 29mentioning

confidence: 68%

Construction of positive definite cubature formulae and approximation of functions via Voronoi tessellations

Guessab

Schmeißer

2008

Adv Comput Math

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“…We also show some upper bounds on G n that were obtained in [3], [5] by evaluating the mean squared error of certain lattices when used to quantize uniformly distributed inputs. The following lattices were used: the so-called Voronoi lattices of the first type, A n * (the subscript indicates the dimension, and the asterisk that these are the duals of the root lattices A n -see [3]); the lattices D n * ; and, in the column of the It is known that A 1 * (the integers) and A 2 * (the familiar two-dimensional hexagonal lattice) are optimal for quantizing uniformly distributed inputs in one and two dimensions ([8], [10], [18]), and that the body-centered cubic lattice A 3 * is similarly optimal among lattice quantizers in three dimensions [1]. It can be seen from the Gosset's (see [4], [5], [15], [16], [19], [24]), and the existence of these two lattices explains the small gaps in the [14]; see also [21].…”

Section: Introductionmentioning

confidence: 99%

A lower bound on the average error of vector quantizers (Corresp.)

Conway

Sloane

1985

IEEE Trans. Inform. Theory

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A lower bound is proposed for the mean squared error of an n-dimensional vector quantizer with a large number of output points. Although no formal proof has been found, a plausible geometrical argument is given for believing that the bound is correct. The new bound is analogous to the Rogers bound for packing spheres, the Coxeter-Few-Rogers bound for covering space with spheres, and the Coxeter-Bo. . ro. . czky bound for packing spherical caps. It is significantly stronger than Zador's sphere bound for quantizers.

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“…German location theorists, Losch and Christaller, claimed already in the thirties that if only the transportation cost matters (ij/ = 0) then of all arrangements in the plane the regular hexagonal arrangement of the market areas is optimal (see [1,7]). This has been shown in a certain sense by Fejes T6th [2] in 1953, whose basic theorem is given below. Similar problems have been investigated by Steinhaus [8], Fejes T6th [3,4], and Heppes and Sziisz [5].…”

Section: Introductionmentioning

confidence: 79%

“…We shall make use of the following corollary of Fejes T6th's theorem, which can be read out of the proof in [2].…”

Section: Bounds For the Function Tmentioning

confidence: 99%