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.
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 .
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.
- 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 ?