In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suitable 3-term
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
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 to each other under some inner product.
The most widely used orthogonal polynomials are the class ...
. 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
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 ...
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
:
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.
See also
*
Jacobi operator
A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix. It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel me ...
References
* Reprinted by Dover 2011,
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*{{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