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Biorthogonal polynomial
In mathematics, a biorthogonal polynomial is a polynomial that is orthogonal to several different measures. Biorthogonal polynomials are a generalization of orthogonal polynomials and share many of their properties. There are two different concepts of biorthogonal polynomials in the literature: Iserles & Nørsett (1988) introduced the concept of polynomials biorthogonal with respect to a sequence of measures, while Szegő introduced the concept of two sequences of polynomials that are biorthogonal with respect to each other.
Polynomials biorthogonal with respect to a sequence of measures
A polynomial p is called biorthogonal with respect to a sequence of measures \( \mu_1, \mu_2, ... \) if
\( \int p(x) \, d\mu_i(x) =0 \) whenever \( i ≤ deg(p). \)
Biorthogonal pairs of sequences
Two sequences \( \psi_0, \psi_1, ... \) and \( \phi_0, \phi_1, ... \) of polynomials are called biorthogonal (for some measure μ) if
\( \int \phi_m(x)\psi_n(x) \, d\mu(x) = 0 \)
whenever m ≠ n.
The definition of biorthogonal pairs of sequences is in some sense a special case of the definition of biorthogonality with respect to a sequence of measures. More precisely two sequences \( \psi_0, \psi_1, ...\) and \( \phi_0, \phi_1, ... \) of polynomials are biorthogonal for the measure μ if and only if the sequence \(\psi_0, \psi_1, ... \) is biorthogonal for the sequence of measures \( \phi_0μ, \phi_1\mu, ...,\) and the sequence \(\phi_0, \phi_1, ...\) is biorthogonal for the sequence of measures \( \psi_0\mu, \psi_1\mu,...\).
References
Iserles, Arieh; Nørsett, Syvert Paul (1988), "On the theory of biorthogonal polynomials", Transactions of the American Mathematical Society 306 (2): 455–474, doi:10.2307/2000806, ISSN 0002-9947, MR 933301
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