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 μ₁, μ₂, ... if

\int p(x)\,d\mu _{i}(x)=0

whenever i ≤ deg(p).

Biorthogonal pairs of sequences

Two sequences ψ₀, ψ₁, ... and φ₀, φ₁, ... 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 ψ₀, ψ₁, ... and φ₀, φ₁, ... of polynomials are biorthogonal for the measure μ if and only if the sequence ψ₀, ψ₁, ... is biorthogonal for the sequence of measures φ₀μ, φ₁μ, ..., and the sequence φ₀, φ₁, ... is biorthogonal for the sequence of measures ψ₀μ, ψ₁μ,....

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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