Sobolev-type orthogonal polynomials and their zeros
When σ is a quasi-definite moment functional on P, the space of polynomials in one variable, we consider a symmetric bilinear form φ(·, ·) on P ×P defined by φ(p, q) := <σ, pq> +λp^(r)(a)q^(r)(a)+µp^(s)(b)q^(s)(b), where λ, µ, a, b are complex numbers and r, s are non-negative integers. We fin...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Sapienza Università Editrice
1997-01-01
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| Series: | Rendiconti di Matematica e delle Sue Applicazioni |
| Subjects: | |
| Online Access: | https://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/1997(3)/423-444.pdf |
| _version_ | 1811332484767416320 |
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| author | D.H. Kim K.H. Kwon F. Marcellán S.B. Park |
| author_facet | D.H. Kim K.H. Kwon F. Marcellán S.B. Park |
| author_sort | D.H. Kim |
| collection | DOAJ |
| description | When σ is a quasi-definite moment functional on P, the space of polynomials in one variable, we consider a symmetric bilinear form φ(·, ·) on P ×P defined by φ(p, q) := <σ, pq> +λp^(r)(a)q^(r)(a)+µp^(s)(b)q^(s)(b), where λ, µ, a, b are complex numbers and r, s are non-negative integers. We find a necessary and sufficient condition under which there is an orthogonal polynomial system {R_n(x)}^∞_{n=0} relative to φ and discuss their algebraic properties. When σ is semi-classical, we show that {R_n(x)}^∞_{n=0} must satisfy a second order differential equation with polynomial coefficients. When σ is positive-definite and λ, µ, a, b are real, we investigate the relations between zeros of {R_n(x)}^∞_{n=0} and of the system of the orthogonal polinomiels relative to σ. |
| first_indexed | 2024-04-13T16:37:40Z |
| format | Article |
| id | doaj.art-f42b5dcfb50e47b3bb445d6f0bf6473f |
| institution | Directory Open Access Journal |
| issn | 1120-7183 2532-3350 |
| language | English |
| last_indexed | 2024-04-13T16:37:40Z |
| publishDate | 1997-01-01 |
| publisher | Sapienza Università Editrice |
| record_format | Article |
| series | Rendiconti di Matematica e delle Sue Applicazioni |
| spelling | doaj.art-f42b5dcfb50e47b3bb445d6f0bf6473f2022-12-22T02:39:23ZengSapienza Università EditriceRendiconti di Matematica e delle Sue Applicazioni1120-71832532-33501997-01-01173423444Sobolev-type orthogonal polynomials and their zerosD.H. Kim0K.H. Kwon1F. Marcellán2S.B. Park3Department of Mathematics KaistDepartment of Mathematics KaistUniversidad Carlos III de MadridDepartment of Mathematics KaistWhen σ is a quasi-definite moment functional on P, the space of polynomials in one variable, we consider a symmetric bilinear form φ(·, ·) on P ×P defined by φ(p, q) := <σ, pq> +λp^(r)(a)q^(r)(a)+µp^(s)(b)q^(s)(b), where λ, µ, a, b are complex numbers and r, s are non-negative integers. We find a necessary and sufficient condition under which there is an orthogonal polynomial system {R_n(x)}^∞_{n=0} relative to φ and discuss their algebraic properties. When σ is semi-classical, we show that {R_n(x)}^∞_{n=0} must satisfy a second order differential equation with polynomial coefficients. When σ is positive-definite and λ, µ, a, b are real, we investigate the relations between zeros of {R_n(x)}^∞_{n=0} and of the system of the orthogonal polinomiels relative to σ.https://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/1997(3)/423-444.pdfsobolev-type orthogonal polynomialsquasi-definite moment functionalsdifferential equationszeros |
| spellingShingle | D.H. Kim K.H. Kwon F. Marcellán S.B. Park Sobolev-type orthogonal polynomials and their zeros Rendiconti di Matematica e delle Sue Applicazioni sobolev-type orthogonal polynomials quasi-definite moment functionals differential equations zeros |
| title | Sobolev-type orthogonal polynomials and their zeros |
| title_full | Sobolev-type orthogonal polynomials and their zeros |
| title_fullStr | Sobolev-type orthogonal polynomials and their zeros |
| title_full_unstemmed | Sobolev-type orthogonal polynomials and their zeros |
| title_short | Sobolev-type orthogonal polynomials and their zeros |
| title_sort | sobolev type orthogonal polynomials and their zeros |
| topic | sobolev-type orthogonal polynomials quasi-definite moment functionals differential equations zeros |
| url | https://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/1997(3)/423-444.pdf |
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