Exercise 12, § 2

Functions of a Real Variable, VI, 295

For every polynomial Q with real coefficients such that Q(0) = Q(1) one writes $\tilde{Q}$ for the continuous function on $\mathbf{R}$ such that $\tilde{Q}(x) = Q(x)$ for $0 \leqslant x \leqslant 1$ and $\tilde{Q}(x + 1) = \tilde{Q}(x)$ for every $x \in \mathbf{R}$ .

a) For every integer $m \geqslant 2$ the Bernoulli polynomial $B_m$ is the unique polynomial Q with real coefficients, of degree m, monic, and such that $\int_0^1 Q(x)\,dx = 0$ , $Q(1) = Q(0)$ and such that the function $\tilde{Q}$ is $m - 2$ times differentiable on $\mathbf{R}$ and has a continuous $(m - 2)^{nd}$ derivative. (Argue by recursion on m.)

b) Show that for $m \geqslant 1$

\int_0^1 B_m(x)\,e^{-2\pi i n x}\,dx = \begin{cases} 0 & \text{if } n = 0 \\ -\frac{m!}{(2\pi i n)^m} & \text{if } n \neq 0 \text{ on } \mathbf{Z}. \end{cases}

*c) Deduce from b) that, for $0 \leqslant x \leqslant 1$ and $m \geqslant 2$ one has

B_m(x) = -\frac{m!}{(2\pi i)^m} \sum_{n \in \mathbf{Z} - \{0\}} \frac{e^{2\pi i n x}}{n^m}

the series being normally convergent.

Solution

No solution has been written yet.