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<p>We develop a theory of polynomials and, in particular, an analog of the theory of Legendre orthogonal polynomials on the bubble-diamond fractals, a class of fractal sets that can be viewed as the completion of a limit of a sequence of finite graph approximations. In this setting, a polynomial of degree <i>j</i> can be viewed as a multiharmonic function, a solution of the equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1782_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Delta ^{j+1}u=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We prove that the sequence of orthogonal polynomials we construct obeys a three-term recursion formula.</p>

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Orthogonal Polynomials on Bubble-Diamond Fractals

  • 期刊论文

  • 发表日期：

    2025年8月9日

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  • Elena Axinn,
  • Calvin Osborne,
  • Kasso A. Okoudjou,
  • Olivia Rigatti,
  • Helen Shi

摘要

We develop a theory of polynomials and, in particular, an analog of the theory of Legendre orthogonal polynomials on the bubble-diamond fractals, a class of fractal sets that can be viewed as the completion of a limit of a sequence of finite graph approximations. In this setting, a polynomial of degree j can be viewed as a multiharmonic function, a solution of the equation \(\Delta ^{j+1}u=0\) Δ j + 1 u = 0 . We prove that the sequence of orthogonal polynomials we construct obeys a three-term recursion formula.