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Favard's theorem

In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suitable three-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 fractions many years before Favard's paper, and was rediscovered several times by other authors before Favard's work.

Statement

Suppose that {y_0, y_1, \dots } is a sequence of polynomials, where y_n has degree n and y_0 = 1. If this is a sequence of orthogonal polynomials for some positive weight function then it satisfies a three-term recurrence relation. Favard's theorem is roughly a converse of this, and states that if these polynomials satisfy a three-term recurrence relation of the form : y_{n+1}= (x-c_n)y_n - d_n y_{n-1} for some numbers c_n and d_n, then the polynomials y_n form an orthogonal sequence for some linear functional \Lambda with \Lambda(1)=1; in other words \Lambda(y_m y_n)=0 if m \neq n.

The linear functional \Lambda is unique, and is given by \Lambda(1)=1, \Lambda(y_n)=0 if n0.

The functional \Lambda satisfies \Lambda(y_n^2)=d_n , \Lambda(y^2_{n-1}), which implies that \Lambda 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,

Wikipedia Source

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orthogonal-polynomials theorems-in-approximation-theory

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