Biorthogonal Polynomials

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: 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 ''μ''₁, ''μ''₂, ... if
:$\int p(x) \, d\mu_i(x) =0$ whenever ''i'' ≤ deg(''p'').

Biorthogonal pairs of sequences

Two sequences ''ψ''₀, ''ψ''₁, ... and ''φ''₀, ''φ''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 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

*{{Citation , last1=Iserles , first1=Arieh , last2=Nørsett , first2=Syvert Paul , title=On the theory of biorthogonal polynomials , doi=10.2307/2000806 , mr=933301 , year=1988 , journal=Transactions of the American Mathematical Society , issn=0002-9947 , volume=306 , issue=2 , pages=455–474, jstor=2000806 , doi-access=free

Orthogonal polynomials