The Christoffel function is the reciprocal of the diagonal Christoffel-Darboux kernel,
![lambda_n(x;mu)=1/(K_n(x,x;mu))=(sum_(j=0)^n[p_j(x)]^2)^(-1),](/images/equations/ChristoffelFunction/NumberedEquation1.svg) |
(1)
|
where 
are orthonormal polynomials
for a finite positive measure 
on the real line with finite
moments of all orders and infinite support. Here 
is real.
This convention uses polynomials of polynomial
degree at most 
;
another common convention indexes the same function by 
, the number of terms.
The function has the variational characterization
![lambda_n(x;mu)=min_(q(x)=1,degq<=n)int_R[q(t)]^2dmu(t),](/images/equations/ChristoffelFunction/NumberedEquation2.svg) |
(2)
|
where the minimum is over real polynomials. Its unique minimizer is
 |
(3)
|
Expanding 
in the orthonormal polynomials
and applying the Schwarz's inequality to 
proves the characterization (Simon
2008).
The Christoffel function is positive and nonincreasing as 
increases. At the zeros 
of 
, the weights in Gaussian
quadrature are the Christoffel numbers
 |
(4)
|
Scaling the measure by 
multiplies the Christoffel function by 
.
For 
on ![[-1,1]](/images/equations/ChristoffelFunction/Inline14.svg)
,
the Legendre polynomials give
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
The figure compares 
, 1, 2, 4, and 8. The marked points on the blue curve 
are at 
, 0, and 
, with heights 
, 
, and 
. These are the weights of three-point
Gaussian quadrature.
