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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Bibliographic Details
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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Summary: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>.
ISSN:2075-1680

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