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 Favard (1935) and Shohat (1938), 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.

Suppose that { y 0 , y 1 , … } {\displaystyle \{y_{0},y_{1},\dots \}} is a sequence of polynomials, where y n {\displaystyle y_{n}} has degree n {\displaystyle n} and y 0 = 1 {\displaystyle 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 {\displaystyle y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}}

for some numbers c n {\displaystyle c_{n}} and d n {\displaystyle d_{n}} , then the polynomials y n {\displaystyle y_{n}} form an orthogonal sequence for some linear functional Λ {\displaystyle \Lambda } with Λ ( 1 ) = 1 {\displaystyle \Lambda (1)=1} ; in other words Λ ( y m y n ) = 0 {\displaystyle \Lambda (y_{m}y_{n})=0} if m ≠ n {\displaystyle m\neq n} .

The linear functional Λ {\displaystyle \Lambda } is unique, and is given by Λ ( 1 ) = 1 {\displaystyle \Lambda (1)=1} , Λ ( y n ) = 0 {\displaystyle \Lambda (y_{n})=0} if n > 0 {\displaystyle n>0} .

The functional Λ {\displaystyle \Lambda } satisfies Λ ( y n 2 ) = d n Λ ( y n − 1 2 ) {\displaystyle \Lambda (y_{n}^{2})=d_{n}\,\Lambda (y_{n-1}^{2})} , which implies that Λ {\displaystyle \Lambda } is positive definite if (and only if) the numbers c n {\displaystyle c_{n}} are real and the numbers d n {\displaystyle d_{n}} are positive.

• Jacobi operator
• James Alexander Shohat
• Jean Favard

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