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 orthogonality, orthogonal to each other under some inner product.
The most widely used orthogonal polynomial ...
. 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 (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
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 ...
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