In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suitable

three-term recurrence relation In mathematics, and especially in numerical analysis, a homogeneous linear three-term recurrence relation (TTRR, the qualifiers "homogeneous linear" are usually taken for granted) is a recurrence relation of the form :y_=a_n y_n + b_n y_ for n=1,2, ...

is a sequence of

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geom ...

. The theorem was introduced in the theory of orthogonal polynomials by and , though essentially the same theorem was used by Stieltjes in the theory of

continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...

s many years before Favard's paper, and was rediscovered several times by other authors before Favard's work.

Statement

Suppose that ''y''₀ = 1, ''y''₁, ... is a sequence of polynomials where ''y''_''n'' has degree ''n''. If this is a sequence of orthogonal polynomials for some positive weight function then it satisfies a 3-term recurrence relation. Favard's theorem is roughly a converse of this, and states that if these polynomials satisfy a 3-term recurrence relation of the form :

y_= (x-c_n)y_n - d_n y_

for some numbers ''c''_''n'' and ''d''_''n'', then the polynomials ''y''_''n'' form an orthogonal sequence for some linear functional Λ with Λ(1)=1; in other words Λ(''y''_''m''''y''_''n'') = 0 if ''m'' ≠ ''n''. The linear functional Λ is unique, and is given by Λ(1) = 1, Λ(''y''_''n'') = 0 if ''n'' > 0. The functional Λ satisfies Λ(''y'') = ''d''_''n'' Λ(''y''), which implies that Λ is positive definite if (and only if) the numbers ''c''_''n'' are real and the numbers ''d''_''n'' are positive.

References

* Reprinted by Dover 2011, * * * *{{Citation , last1=Shohat , first1=J. , title=Sur les polynômes orthogonaux généralises. , language=French , zbl=0019.40503 , year=1938 , journal=C. R. Acad. Sci. Paris , volume=207 , pages=556–558 Orthogonal polynomials Theorems in approximation theory

Statement

See also

References