Favard's theorem
Encyclopedia
In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suitable 3-term recurrence relation is a sequence of orthogonal polynomials
. 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
s many years before Favard's paper, and was rediscovered several times by other authors before Favard's work.
for some numbers cn and dn,
then the polynomials yn form an orthogonal sequence for some linear function Λ with Λ(1)=1; in other words Λ(ymyn) = 0 if m ≠ n.
The linear functional Λ is unique, and is given by Λ(1) = 1, Λ(yn) = 0 if n > 0.
The functional Λ satisfies Λ(y) = dn Λ(y), which implies that Λ is positive definite if (and only if) the numbers cn are real and the numbers dn are positive.
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
. The theorem was introduced in the theory of orthogonal polynomials by and , though essentially the same theorem was used by Stieltjes
Thomas Joannes Stieltjes
Thomas Joannes Stieltjes was a Dutch mathematician. He was born in Zwolle and died in Toulouse, France. He was a pioneer in the field of moment problems and contributed to the study of continued fractions....
in the theory of continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
s many years before Favard's paper, and was rediscovered several times by other authors before Favard's work.
Statement
Suppose that y0 = 1, y1, ... is a sequence of polynomials where yn 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 formfor some numbers cn and dn,
then the polynomials yn form an orthogonal sequence for some linear function Λ with Λ(1)=1; in other words Λ(ymyn) = 0 if m ≠ n.
The linear functional Λ is unique, and is given by Λ(1) = 1, Λ(yn) = 0 if n > 0.
The functional Λ satisfies Λ(y) = dn Λ(y), which implies that Λ is positive definite if (and only if) the numbers cn are real and the numbers dn are positive.