Favard's theorem characterizes orthogonal polynomials on the real line by a three-term recurrence
relation. Let 
and 
,
and define monic polynomials by
 |
(1)
|
for 
,
1, ..., where 
is real, 
for 
, and 
. There exists a finite positive
measure 
on the real line, not concentrated on any finite set
and having finite moments of every order, such that
 |
(2)
|
where 
and 
is the Kronecker delta. Conversely, monic
polynomials orthogonal with respect to
such a measure satisfy this recurrence
relation (Favard 1935).
The coefficients satisfy 
, so the squared norms
are
 |
(3)
|
The measure need not have a density and need not be unique even after its total mass is fixed. This is related to the moment
problem. With 
,
the recurrence relation is encoded by a Jacobi matrix. The theorem concerns orthogonal
polynomials in general, rather than only the classical families that satisfy
second-order differential equations.
