On the theory of biorthogonal polynomials
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- by A. Iserles and S. P. Nørsett PDF
- Trans. Amer. Math. Soc. 306 (1988), 455-474 Request permission
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 \[ \int _{ - \infty }^\infty {{p_m}(x) d\varphi (x,{\mu _l}) = 0,} \qquad l = 1, 2, \ldots , m,\;m \geqslant 1\] , 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.References
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Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 306 (1988), 455-474
- MSC: Primary 42C05; Secondary 33A65
- DOI: https://doi.org/10.1090/S0002-9947-1988-0933301-8
- MathSciNet review: 933301