On means of distances on the surface of a sphere. II. (Upper bounds)

Wagner, Gerold

doi:10.2140/pjm.1992.154.381

Cited by 51 publications

(27 citation statements)

References 2 publications

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“…The asymptotics were studied by G. Wagner [23], [24] for the case 0 < s < d (and also for s < 0, but we will not consider such values here). For 0 < s < d, the energy integral…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Asymptotics for minimal discrete energy on the sphere

Kuijlaars

Saff

1998

Trans. Amer. Math. Soc.

161

124

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Abstract. We investigate the energy of arrangements of N points on the surface of the unit sphere S d in R d+1 that interact through a power law potential V = 1/r s , where s > 0 and r is Euclidean distance. With E d (s, N ) denoting the minimal energy for such N -point arrangements we obtain bounds (valid for all N ) for E d (s, N ) in the cases when 0 < s < d and 2 ≤ d < s. For s = d, we determine the precise asymptotic behavior ofAs a corollary, lower bounds are given for the separation of any pair of points in an N -point minimal energy configuration, when s ≥ d ≥ 2.For the unit sphere in R 3 (d = 2), we present two conjectures concerning the asymptotic expansion of E 2 (s, N ) that relate to the zeta function ζ L (s) for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the first of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of ζ L (s) when 0 < s < 2 (the divergent case).

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“…The asymptotics were studied by G. Wagner [23], [24] for the case 0 < s < d (and also for s < 0, but we will not consider such values here). For 0 < s < d, the energy integral…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…(Here and in the following, C denotes a positive constant that may depend on s and d, but not on N . ) Wagner [24] also derived an upper bound for the case d = 2:…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Asymptotics for minimal discrete energy on the sphere

Kuijlaars

Saff

1998

Trans. Amer. Math. Soc.

161

124

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“…The beamforming vectors have unit norm due to transmit power constraint, and thus the transmitter requires a spherical interpolation algorithm. Some spherical averaging methods and their appli-DRAFT cations to spherical interpolation have been proposed [16]- [19]. However, it is not easy to directly use the algorithms to interpolate beamforming vectors, because the optimal beamforming vector is not unique and it is difficult to determine the coefficients for higher order interpolation.…”

Section: Introductionmentioning

confidence: 99%

Interpolation based transmit beamforming for MIMO-OFDM with limited feedback

Choi

Heath

2004

2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577)

119

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Transmit beamforming with receive combining is a simple method for exploiting the significant diversity provided by multiple-input multiple-output (MIMO) systems, and the use of orthogonal frequency division multiplexing (OFDM) enables low complexity implementation of this scheme over frequency selective MIMO channels. Optimal beamforming requires channel state information in the form of the beamforming vectors corresponding to all the OFDM subcarriers. In an attempt to reduce the amount of feedback information, we propose a new approach to transmit beamforming that combines partial feedback and beamformer interpolation. In the proposed architecture, the receiver sends a fraction of information about optimal beamforming vectors to the transmitter, and the transmitter computes the beamforming vectors for all subcarriers through modified spherical linear interpolation of the conveyed beamforming vectors. Simulation results show that the proposed beamforming method requires much less feedback information than optimal beamforming while it exhibits slight diversity loss compared to the latter.

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“…Our construction of the Chebyshev-type cubature for the sphere is very similar to a construction of Wagner [27] which he used in his work on a problem in potential theory (in a manner somewhat similar to that of the application in [8]). Despite this similarity, we chose to give a full proof of it here since some parts in Wagner's construction (such as the exact partition of the sphere) are only sketched and since our construction gives explicit bounds on the number of nodes in the cubature (at the expense of getting only an approximate cubature formula), whereas his only shows existence.…”

Section: Application To Construction Of Local Cubaturesmentioning

confidence: 95%

Simple universal bounds for Chebyshev-type quadratures

Peled

2010

Journal of Approximation Theory

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A Chebyshev-type quadrature for a probability measure σ is a distribution which is uniform on n points and has the same first k moments as σ . We give an upper bound for the minimal n required to achieve a given degree k, for σ supported on an interval. In contrast to previous results of this type, our bound uses only simple properties of σ and is applicable in wide generality. We also obtain a lower bound for the required number of nodes which only uses estimates on the moments of σ . Examples illustrating the sharpness of our bounds are given. As a corollary of our results, we obtain an apparently new result on the Gaussian quadrature.In addition, we suggest another approach to bounding the minimal number of nodes required in a Chebyshev-type quadrature, utilizing a random choice of the nodes, and propose the challenge of analyzing its performance. A preliminary result in this direction is proved for the uniform measure on the cube. Finally, we apply our bounds to the construction of point sets on the sphere and cylinder which form local approximate Chebyshev-type quadratures. These results were needed recently in the context of understanding how well a Poisson process can approximate certain continuous distributions. The paper concludes with a list of open questions.

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On means of distances on the surface of a sphere. II. (Upper bounds)

Cited by 51 publications

References 2 publications

Asymptotics for minimal discrete energy on the sphere

Asymptotics for minimal discrete energy on the sphere

Interpolation based transmit beamforming for MIMO-OFDM with limited feedback

Simple universal bounds for Chebyshev-type quadratures

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