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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Sobolev orthogonal polynomials and spectral differential equations

Authors: I. H. Jung, K. H. Kwon, D. W. Lee and L. L. Littlejohn
Journal: Trans. Amer. Math. Soc. 347 (1995), 3629-3643
MSC: Primary 34L05; Secondary 33C45
MathSciNet review: 1308015
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Abstract: We find necessary and sufficient conditions for a spectral differential equation

$\displaystyle {L_N}[y](x) = \sum\limits_{i = 1}^N {{\ell _i}(x){y^{(i)}}(x) = {\lambda _n}y(x)} $

to have Sobolev orthogonal polynomials of solutions, which are orthogonal relative to the Sobolev (pseudo-) inner product

$\displaystyle \phi (p,q) = \int_\mathbb{R}^{} {pqd\mu + \int_\mathbb{R}^{} {p'q'dv,} } $

where $ d\mu $ and $ dv$ are signed Borel measures having finite moments. This result generalizes a result by H. L. Krall, which handles the case when $ dv = 0$.

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Keywords: Sobolev orthogonal polynomials, Sobolev bilinear forms, spectral differential equations
Article copyright: © Copyright 1995 American Mathematical Society