The minimizers of the p-frame potential

Zhiqiang Xu; Zili Xu

doi:10.1016/j.acha.2020.04.003

The minimizers of the p-frame potential

Zhiqiang Xu
, Zili Xu^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

For any positive real number p, the p-frame potential of N unit vectors X:={x₁,…,x_N}⊂R^d is defined as FP_p,N,d(X)=∑_i≠j|〈x_i,x_j〉|^p. In this paper, we focus on this quantity for N=d+1 points and establish uniqueness of minimizers of FP_p,d+1,d for all p∈(0,2). Our results completely solve the minimization problem of the p-frame potential when N=d+1, confirming a conjecture posed by Chen, Gonzales, Goodman, Kang and Okoudjou [4].

Original language	English
Pages (from-to)	366-379
Number of pages	14
Journal	Applied and Computational Harmonic Analysis
Volume	52
DOIs	https://doi.org/10.1016/j.acha.2020.04.003
State	Published - May 2021
Externally published	Yes

Keywords

Frame potential
Spherical designs
Tight frames

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10.1016/j.acha.2020.04.003

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title = "The minimizers of the p-frame potential",

abstract = "For any positive real number p, the p-frame potential of N unit vectors X:=\{x1,…,xN\}⊂Rd is defined as FPp,N,d(X)=∑i≠j|〈xi,xj〉|p. In this paper, we focus on this quantity for N=d+1 points and establish uniqueness of minimizers of FPp,d+1,d for all p∈(0,2). Our results completely solve the minimization problem of the p-frame potential when N=d+1, confirming a conjecture posed by Chen, Gonzales, Goodman, Kang and Okoudjou [4].",

keywords = "Frame potential, Spherical designs, Tight frames",

author = "Zhiqiang Xu and Zili Xu",

note = "Publisher Copyright: {\textcopyright} 2020 Elsevier Inc.",

year = "2021",

month = may,

doi = "10.1016/j.acha.2020.04.003",

language = "英语",

volume = "52",

pages = "366--379",

journal = "Applied and Computational Harmonic Analysis",

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T1 - The minimizers of the p-frame potential

AU - Xu, Zhiqiang

AU - Xu, Zili

PY - 2021/5

Y1 - 2021/5

N2 - For any positive real number p, the p-frame potential of N unit vectors X:={x1,…,xN}⊂Rd is defined as FPp,N,d(X)=∑i≠j|〈xi,xj〉|p. In this paper, we focus on this quantity for N=d+1 points and establish uniqueness of minimizers of FPp,d+1,d for all p∈(0,2). Our results completely solve the minimization problem of the p-frame potential when N=d+1, confirming a conjecture posed by Chen, Gonzales, Goodman, Kang and Okoudjou [4].

AB - For any positive real number p, the p-frame potential of N unit vectors X:={x1,…,xN}⊂Rd is defined as FPp,N,d(X)=∑i≠j|〈xi,xj〉|p. In this paper, we focus on this quantity for N=d+1 points and establish uniqueness of minimizers of FPp,d+1,d for all p∈(0,2). Our results completely solve the minimization problem of the p-frame potential when N=d+1, confirming a conjecture posed by Chen, Gonzales, Goodman, Kang and Okoudjou [4].

KW - Frame potential

KW - Spherical designs

KW - Tight frames

UR - https://www.scopus.com/pages/publications/85084393546

U2 - 10.1016/j.acha.2020.04.003

DO - 10.1016/j.acha.2020.04.003

M3 - 文章

AN - SCOPUS:85084393546

SN - 1063-5203

VL - 52

SP - 366

EP - 379

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

ER -

The minimizers of the p-frame potential

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Keywords

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