Scholarlink

<p>We derive universal lower and upper bounds for max–min and min–max problems (also known as polarization) for the potential of spherical (<i>k</i>,&#xa0;<i>k</i>)-designs and provide certain examples, including unit-norm tight frames, that attain these bounds. The universality is understood in the sense that the bounds hold for all spherical (<i>k</i>,&#xa0;<i>k</i>)-designs and for a large class of potential functions, and the bounds involve certain nodes and weights that are independent of the potential. When the potential function is <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1659_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">\(h(t)=t^{2k}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>t</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, we prove an optimality property of the spherical (<i>k</i>,&#xa0;<i>k</i>)-designs in the class of all spherical codes of the same cardinality both for max–min and min–max polarization problems.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Bounds on discrete potentials of spherical (k, k)-designs

  • 期刊论文

  • 发表日期：

    2025年6月10日

查看全文
  • S. V. Borodachov,
  • P. G. Boyvalenkov,
  • P. D. Dragnev,
  • D. P. Hardin,
  • E. B. Saff,
  • M. M. Stoyanova

摘要

We derive universal lower and upper bounds for max–min and min–max problems (also known as polarization) for the potential of spherical (k, k)-designs and provide certain examples, including unit-norm tight frames, that attain these bounds. The universality is understood in the sense that the bounds hold for all spherical (k, k)-designs and for a large class of potential functions, and the bounds involve certain nodes and weights that are independent of the potential. When the potential function is \(h(t)=t^{2k}\) h ( t ) = t 2 k , we prove an optimality property of the spherical (k, k)-designs in the class of all spherical codes of the same cardinality both for max–min and min–max polarization problems.