Krein–Sobolev Orthogonal Polynomials II

Krein–Sobolev Orthogonal Polynomials II

In a recent paper, Littlejohn and Quintero studied the orthogonal polynomials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>K</mi><mi&g...

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Main Authors: Alexander Jones, Lance Littlejohn, Alejandro Quintero Roba
Format: Article
Language:English
Published: MDPI AG 2025-02-01
Series:Axioms
Subjects:
one-dimensional Krein Laplacian self-adjoint operator
left-definite spectral theory
Althammer polynomials
Krein–Sobolev polynomials
Online Access:https://www.mdpi.com/2075-1680/14/2/115
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_version_ 1849720218610302976
author Alexander Jones
Lance Littlejohn
Alejandro Quintero Roba
author_facet Alexander Jones
Lance Littlejohn
Alejandro Quintero Roba
author_sort Alexander Jones
collection DOAJ
description In a recent paper, Littlejohn and Quintero studied the orthogonal polynomials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>K</mi><mi>n</mi></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></semantics></math></inline-formula>—which they named <i>Krein–Sobolev polynomials</i>—that are orthogonal in the classical Sobolev space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mn>1</mn></msup><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></semantics></math></inline-formula> with respect to the (positive-definite) inner product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>,</mo><mi>c</mi></mrow></msub><mo>:</mo><mo>=</mo><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mfenced separators="" open="(" close=")"><mi>f</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>−</mo><mi>f</mi><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mfenced><mfenced separators="" open="(" close=")"><mover><mi>g</mi><mo>¯</mo></mover><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>−</mo><mover><mi>g</mi><mo>¯</mo></mover><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mfenced></mrow><mn>2</mn></mfrac></mstyle><mo>+</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mn>1</mn></mrow><mn>1</mn></msubsup><mrow><mo>(</mo><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mover><mi>g</mi><mo>¯</mo></mover><mo>′</mo></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mover><mi>g</mi><mo>¯</mo></mover><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>d</mi><mi>x</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <i>c</i> is a fixed, positive constant. These polynomials generalize the Althammer (or Legendre–Sobolev) polynomials first studied by Althammer and Schäfke. The Krein–Sobolev polynomials were found as a result of a left-definite spectral study of the self-adjoint Krein Laplacian operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">K</mi><mi>c</mi></msub></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>c</mi><mo>></mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> Other than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mn>0</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>K</mi><mn>1</mn></msub><mo>,</mo></mrow></semantics></math></inline-formula> these polynomials are not eigenfunctions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">K</mi><mi>c</mi></msub><mo>.</mo></mrow></semantics></math></inline-formula> As shown by Littlejohn and Quintero, the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>K</mi><mi>n</mi></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></semantics></math></inline-formula> forms a complete orthogonal set in the first left-definite space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>H</mi><mn>1</mn></msup><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>,</mo><msub><mrow><mo>(</mo><mo>·</mo><mo>,</mo><mo>·</mo><mo>)</mo></mrow><mrow><mn>1</mn><mo>,</mo><mi>c</mi></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> associated with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi mathvariant="script">K</mi><mi>c</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><mrow><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Furthermore, they show that, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>K</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> has <i>n</i> distinct zeros in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> In this note, we find an explicit formula for Krein–Sobolev polynomials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>K</mi><mi>n</mi></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></semantics></math></inline-formula>.
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spelling doaj-art-ddf8979a1ff940bfa286ec2aefe05db42025-08-20T03:11:58ZengMDPI AGAxioms2075-16802025-02-0114211510.3390/axioms14020115Krein–Sobolev Orthogonal Polynomials IIAlexander Jones0Lance Littlejohn1Alejandro Quintero Roba2Department of Mathematics, Baylor University, Sid Richardson Science Building, 1410 S. 4th Street, Waco, TX 76706, USADepartment of Mathematics, Baylor University, Sid Richardson Science Building, 1410 S. 4th Street, Waco, TX 76706, USADepartment of Mathematics, Baylor University, Sid Richardson Science Building, 1410 S. 4th Street, Waco, TX 76706, USAIn a recent paper, Littlejohn and Quintero studied the orthogonal polynomials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>K</mi><mi>n</mi></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></semantics></math></inline-formula>—which they named <i>Krein–Sobolev polynomials</i>—that are orthogonal in the classical Sobolev space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mn>1</mn></msup><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></semantics></math></inline-formula> with respect to the (positive-definite) inner product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>,</mo><mi>c</mi></mrow></msub><mo>:</mo><mo>=</mo><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mfenced separators="" open="(" close=")"><mi>f</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>−</mo><mi>f</mi><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mfenced><mfenced separators="" open="(" close=")"><mover><mi>g</mi><mo>¯</mo></mover><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>−</mo><mover><mi>g</mi><mo>¯</mo></mover><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mfenced></mrow><mn>2</mn></mfrac></mstyle><mo>+</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mn>1</mn></mrow><mn>1</mn></msubsup><mrow><mo>(</mo><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mover><mi>g</mi><mo>¯</mo></mover><mo>′</mo></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mover><mi>g</mi><mo>¯</mo></mover><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>d</mi><mi>x</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <i>c</i> is a fixed, positive constant. These polynomials generalize the Althammer (or Legendre–Sobolev) polynomials first studied by Althammer and Schäfke. The Krein–Sobolev polynomials were found as a result of a left-definite spectral study of the self-adjoint Krein Laplacian operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">K</mi><mi>c</mi></msub></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>c</mi><mo>></mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> Other than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mn>0</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>K</mi><mn>1</mn></msub><mo>,</mo></mrow></semantics></math></inline-formula> these polynomials are not eigenfunctions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">K</mi><mi>c</mi></msub><mo>.</mo></mrow></semantics></math></inline-formula> As shown by Littlejohn and Quintero, the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>K</mi><mi>n</mi></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></semantics></math></inline-formula> forms a complete orthogonal set in the first left-definite space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>H</mi><mn>1</mn></msup><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>,</mo><msub><mrow><mo>(</mo><mo>·</mo><mo>,</mo><mo>·</mo><mo>)</mo></mrow><mrow><mn>1</mn><mo>,</mo><mi>c</mi></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> associated with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi mathvariant="script">K</mi><mi>c</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><mrow><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Furthermore, they show that, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>K</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> has <i>n</i> distinct zeros in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> In this note, we find an explicit formula for Krein–Sobolev polynomials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>K</mi><mi>n</mi></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></semantics></math></inline-formula>.https://www.mdpi.com/2075-1680/14/2/115one-dimensional Krein Laplacian self-adjoint operatorleft-definite spectral theoryAlthammer polynomialsKrein–Sobolev polynomials
spellingShingle Alexander Jones
Lance Littlejohn
Alejandro Quintero Roba
Krein–Sobolev Orthogonal Polynomials II
Axioms
one-dimensional Krein Laplacian self-adjoint operator
left-definite spectral theory
Althammer polynomials
Krein–Sobolev polynomials
title Krein–Sobolev Orthogonal Polynomials II
title_full Krein–Sobolev Orthogonal Polynomials II
title_fullStr Krein–Sobolev Orthogonal Polynomials II
title_full_unstemmed Krein–Sobolev Orthogonal Polynomials II
title_short Krein–Sobolev Orthogonal Polynomials II
title_sort krein sobolev orthogonal polynomials ii
topic one-dimensional Krein Laplacian self-adjoint operator
left-definite spectral theory
Althammer polynomials
Krein–Sobolev polynomials
url https://www.mdpi.com/2075-1680/14/2/115
work_keys_str_mv AT alexanderjones kreinsobolevorthogonalpolynomialsii
AT lancelittlejohn kreinsobolevorthogonalpolynomialsii
AT alejandroquinteroroba kreinsobolevorthogonalpolynomialsii

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