Biorthogonal polynomial
Open-knowledge reference entry
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
∫
p
(
x
)
d
μ
i
(
x
)
=
0
{\displaystyle \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
∫
ϕ
m
(
x
)
ψ
n
(
x
)
d
μ
(
x
)
=
0
{\displaystyle \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, ...
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