Let
X
=
(
x
1
,
x
2
,
…
,
x
N
)
,
f
:
R
→
C
X = ({x_1},{x_2}, \ldots ,{x_N}),f:{\mathbf {R}} \to {\mathbf {C}}
and let
P
n
{{\mathbf {P}}_n}
be the class of polynomials of degree at most n. The generalized Christoffel function
Λ
n
{\Lambda _n}
corresponding to the measure
d
α
d\alpha
is defined by
\[
Λ
n
(
X
;
f
,
N
,
d
α
)
=
min
π
∈
P
n
−
1
π
(
x
i
)
=
f
(
x
i
)
i
=
1
,
2
,
…
,
N
∫
−
∞
∞
|
π
(
t
)
|
2
d
α
(
t
)
.
{\Lambda _n}(X;f,N,d\alpha ) = \min \limits _{\begin {array}{*{20}{c}} {\pi \in {{\mathbf {P}}_{n - 1}}} \\ {\pi ({x_i}) = f({x_i})} \\ {i = 1,2, \ldots ,N} \\ \end {array} } \int _{ - \infty }^\infty {|\pi (t){|^2}d\alpha (t).}
\]
It is shown that if
α
\alpha
satisfies some rather weak conditions then
lim
n
→
∞
n
Λ
n
(
X
;
f
,
N
,
d
α
)
{\lim _{n \to \infty }}n{\Lambda _n}(X;f,N,d\alpha )
exists and the limit is also evaluated.