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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the theory of biorthogonal polynomials


Authors: A. Iserles and S. P. Nørsett
Journal: Trans. Amer. Math. Soc. 306 (1988), 455-474
MSC: Primary 42C05; Secondary 33A65
MathSciNet review: 933301
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Abstract: Let $ \varphi (x,\,\mu )$ be a distribution in $ x \in {\mathbf{R}}$ for every $ \mu $ in a real parameter set $ \Omega $. Subject to additional technical conditions, we study $ m$th degree monic polynomials $ {p_m}$ that satisfy the biorthogonality conditions

$\displaystyle \int_{ - \infty }^\infty {{p_m}(x)\,d\varphi (x,{\mu _l}) = 0,} \qquad l = 1,\,2, \ldots ,\,m,\;m \geqslant 1$

, for a distinct sequence $ {\mu _1},\,{\mu _2},\, \ldots \; \in \Omega \,$. Necessary and sufficient conditions for existence and uniqueness are established, as well as explicit determinantal and integral representations. We also consider loci of zeros, existence of Rodrigues-type formulae and reducibility to standard orthogonality. The paper is accompanied by several examples of biorthogonal systems.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0933301-8
PII: S 0002-9947(1988)0933301-8
Article copyright: © Copyright 1988 American Mathematical Society