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 |
| Summary: | 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. |
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| ISSN: | 1120-7183 2532-3350 |