The Christoffel-Darboux kernel is the finite sum of products of orthogonal polynomials
 |
(1)
|
where the real polynomials 
have polynomial degree 
and satisfy 
with 
for a positive measure 
on the real line. Here 
is the Kronecker delta.
If 
is the leading coefficient of 
, the Christoffel-Darboux
formula gives
 |
(2)
|
for 
.
The quotient has a removable singularity
at 
,
and the diagonal value is
![K_n(x,x)=(k_n)/(k_(n+1)h_n)[p_(n+1)^'(x)p_n(x)-p_n^'(x)p_(n+1)(x)].](/images/equations/Christoffel-DarbouxKernel/NumberedEquation3.svg) |
(3)
|
For any polynomial 
of polynomial degree
at most 
,
orthogonality gives
 |
(4)
|
Thus the kernel recovers 
from its integrals against the
orthogonal polynomials. It is unchanged
if the individual polynomials are rescaled, since
each 
changes by the square of the same factor (Simon 2008).
At the zeros 
of 
, the weights of Gaussian
quadrature are
 |
(5)
|
The diagonal kernel is positive on the real line, so all these weights are positive. For orthonormal polynomials, exactness of the quadrature gives
 |
(6)
|
for 
,
1, ..., 
.
The weights are Christoffel
numbers.
For the Legendre polynomials, 
and 
, so the first nonconstant kernel is
 |
(7)
|
