The function has a curious property: for any linear function
, and any point
, the integral
evaluates to
. This is easy to check using the fact that odd powers of
integrate to zero:
More generally, for any integer there exists a unique symmetric polynomial
that has degree
in
and
separately and satisfies
for all polynomials
of degree at most
. For example,
(obviously) and
The formula is not really intuitive, and a 3d plot would not help the matter much. To visualize , I plotted
,
, and
below (green, red, blue respectively).



For and large
, the function
approaches the Dirac delta at
, although the convergence is slow, especially when
is close to
. I don’t think there is anything good to be said about the case
.


The existence and uniqueness of are a consequence of the Riesz representation of linear functionals on an inner product space. Indeed, polynomials of degree at most
form such a space
with inner product
, and the functional
is linear for any fixed
. Hence, this functional can be written as
for some
. The function
is a reproducing kernel for this space. Its symmetry is not immediately obvious.
The Legendre polynomials are an orthogonal basis of
; more precisely,
form an orthonormal basis. It’s a general fact about reproducing kernels that
(which, incidentally, proves the symmetry ). Indeed, taking this sum as the definition of
and writing
, we find
This is the Sage code used for the above plots.
n = 20
k = sum([(j+1/2)*legendre_P(j,x)*legendre_P(j,y) for j in range(0,n+1)])
plot(k(x,y=-3/4),(x,-1,1),color='green') + plot(k(x,y=0),(x,-1,1),color='red') + plot(k(x,y=1/2),(x,-1,1),color='blue')
Higher degrees cause some numerical issues…

Post motivated by Math.SE question