We know that
where $B_n(t)$ is Bernoulli polynomials.
My question: Can Bernoulli polynomials be orthogonalized with respect to a weight function $\omega$? or I mean what is a weight function under which the Bernoulli polynomials themselves are orthogonal?
If it is so, what is the weight function $\omega$, how can derive it?
HINT:
We notice that $B_6(t)$ has $2$ real roots and $4$ complex ones. We know that polynomials orthogonal w.r. to some weight concentrated on the real axis have all the roots real.