dual formula to Bernoulli polynomials

Question

$$ \tilde{B}_n(x) = \frac{(-1)^{n + 1}}{n!} \left( \delta^{(n - 1)}(x - 1) - \delta^{(n - 1)}(x) \right) $$

Wikipedia says this formulae is DUAL to the Bernoulli POlynomials but dual in what sense ??

thanks

EDIT: link http://en.wikipedia.org/wiki/Euler_Maclaurin in the section 'derivation by functinal analysis'

Answer 1

As is written in the provided link, the dual base $\{\mathbf{e}^i\}$ to a base $\{\mathbf{e}_i\}$ is defined by the relation $\mathbf{e}^i(\mathbf{e}_j) = \delta_{ij}.$ In the case of Bernoulli polynomials as a base in $L_2([0,1])$ it is $$ \int_0^1 \tilde{B}_i(x) B_j(x)\, dx = \delta_{ij}. $$