For every nonnegative integer there exists a (unique) polynomial
of degree
in
and
separately with the following reproducing property:
for every polynomial of degree at most
, and for every
. For example,
; other examples are found in the post Polynomial delta function.
This fact gives an explicit pointwise bound on a polynomial in terms of its integral on an interval:
where . For example,
.
Although in principle could be any real or complex number, it makes sense to restrict attention to
, where integration takes place. This leads to the search for extreme values of
on the square
. Here is how this function looks for
:



The symmetries are evident here.
Explicitly,
where is the Legendre polynomial of degree
and the factor
is included to make the polynomials an orthonormal set in
. Since
oscillates between
and
, it follows that
and this bound is attained at because
and
.
Is
the minimum value of on the square
? Certainly not for even
. Indeed, differentiating the sum
with respect to and using
, we arrive at
which is negative if is even, ruling out this point as a minimum.
What about odd , then: is it true that
on the square
?
: yes,
is clear enough.
: the inequality
is also true… at least numerically. It can be simplified to
but I do not see a way forward from there. At least on the boundary of
it can be shown without much work:
The quadratic term has no real roots, which is easy to check.
: similar story, the inequality
appears to be true but I can only prove it on the boundary, using
The quartic term has no real roots, which is not so easy to check.
: surprisingly,
which is about
, disproving the conjectural bound
. This fact is not at all obvious from the plot.

Questions:
- Is
on the square
with some universal constant
?
- Is the minimum of
on
always attained on the boundary of
?
- Can
be expressed in closed form, at least for small degrees
?