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2017 Energy optimization for distributions on the sphere and improvement to the Welch bounds
Yan Shuo Tan
Electron. Commun. Probab. 22: 1-12 (2017). DOI: 10.1214/17-ECP73

Abstract

For any Borel probability measure on $\mathbb{R} ^n$, we may define a family of eccentricity tensors. This new notion, together with a tensorization trick, allows us to prove an energy minimization property for rotationally invariant probability measures. We use this theory to give a new proof of the Welch bounds, and to improve upon them for collections of real vectors. In addition, we are able to give elementary proofs for two theorems characterizing probability measures optimizing one-parameter families of energy integrals on the sphere. We are also able to explain why a phase transition occurs for optimizers of these two families.

Citation

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Yan Shuo Tan. "Energy optimization for distributions on the sphere and improvement to the Welch bounds." Electron. Commun. Probab. 22 1 - 12, 2017. https://doi.org/10.1214/17-ECP73

Information

Received: 23 January 2017; Accepted: 7 July 2017; Published: 2017
First available in Project Euclid: 15 August 2017

zbMATH: 1378.60049
MathSciNet: MR3693769
Digital Object Identifier: 10.1214/17-ECP73

Subjects:
Primary: 15A69, 52A40, 60E15

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