Exercise 37, § 1

On keeps the notations of the preceding exercise. (a) Let $f:\mathbf{C}\rightarrow \mathbf{C}$ be an element of $\mathscr{E}_{1/2}$ such that the restriction of $f$ to $\mathbf{R}$ belongs to $L^2(\mathbf{R})$. For every $z\in \mathbf{C}-\mathbf{Z}$, we have

$f(z) =$ sin($\pi \pi z$)$\sum_{n\in\mathbf{Z}}(-1)^n\frac{f(n)}{z-n}$.

(Use the Paley-Wiener theorem and formula a) of Exercise 1.)

(b) Let $f:\mathbf{C}\rightarrow \mathbf{C}$ be an element of $\mathscr{E}_{1/2}$ such that $f$ is bounded on $\mathbf{R}$. For every $z\in \mathbf{C}-\mathbf{Z}$, we have

$f(z) =f(0) +$ sin($\pi \pi z$)$(f'(0) +\sum_{n\in\mathbf{Z}-\{0\}}(-1)^nz\frac{f(n) -f(0)}{n(z-n)})$

(apply the preceding question to the function defined by $z\mapsto (f(z)-f(0))/z$ for $z\not= 0$ and which maps 0 to $f'(0)$).

(c) Let $f$ be an element of $\mathscr{E}_{1/2}$ bounded on $\mathbf{R}$. Then $f'$ is bounded on $\mathbf{R}$ and we have

sup$_{x\in\mathbf{R}}|f'(x)|\leqslant 2$ sup$_{x\in\mathbf{R}}|f(x)|$

(“Bernstein inequality”; reduce to majorizing $|f'(1/2)|$ and apply the preceding formula).

(d) Let $f$ be an element of $\mathscr{E}_{1/2}$ such that the restriction of $f$ to $\mathbf{R}$ belongs to $L^2(\mathbf{R})$. Let $u$ and $v$ be the 1-periodic functions on $\mathbf{R}$ defined by

for $0\leqslant t <1$. For almost every $t\in [-1,1]$, we have

$\widehat{f}(t) = (1- |t|)u(t) + (2i\pi )^{-1}$ s($t$)$v(t)$,

and the coefficients of the Fourier series of $u$ and $v$ are $f(-n)$ and $f'(-n)$, respectively, for $n\in \mathbf{Z}$.

(e) Let $f\in \mathscr{E}_{1/2}$ as in the preceding question. We have for every $z\in \mathbf{C}$ the formula

$f(z) =($ sin($\pi \pi z$))$^2\{(\sum_{n\in\mathbf{Z}}\frac{f(n)}{(z-n)^2}+\sum_{n\in\mathbf{Z}}\frac{f'(n)}{z-n}\}$,

where the series converges symmetrically in $\mathscr{O}(\mathbf{C})$, that is to say, uniformly on the compact subsets of $\mathbf{C}$. (Use the formulas of question f) of Exercise 1.)