Exercise 1, § 3

Let $(a_n)_{n\in\mathbf{N}}$ be a sequence of complex numbers such that there exists a real $A\geqslant 0$ satisfying $n|a_n|\leqslant A$ for every $n\in \mathbf{N}$. For $z\in \mathbf{C}$ of module $<1$, put $\varphi (z) =\sum_{n\geqslant 0}a_nz^n$.

Suppose that there exists $\ell \in \mathbf{C}$ such that

lim$_{xx<\in\mathbf{R}1^+}\varphi (x) =\ell$.

For $t\in \mathbf{R}^*_+$, put

We denote by $dx$ the Lebesgue measure on $\mathbf{R}$ as well as its restriction to $\mathbf{R}_+^*$.

(a) Let $s\leqslant t$ be elements of $\mathbf{R}^*_+$. We have

$|g(s)-g(t)|\leqslant |a_{[t^{-1}+1]}|+ A$ log.

(b) The function $g$ is slowly oscillating for $t\rightarrow 0$.

(c) For every $t\in \mathbf{R}^*_+$, the function $x\mapsto g(t/x)xe^{-x}$ is integrable on $\mathbf{R}_+^*$, with respect to the Haar measure $x^{-1}dx$.

(d) For $0< x <1$, we have

$n\geqslant 0$

(e) Let $f(x) =xe^{-x}$ for $x\in \mathbf{R}_+^*$. We have $f\in L^1(\mathbf{R}^*_+, dx/x)$; its integral is equal to 1, and we have $F =g*f$.

(f) The function $g$ belongs to $L^{\infty}(\mathbf{R}^*_+)$. (Now prove that $F-g$ is bounded by using (a) as well as the assumption.)

(g) The Fourier transform of the function $f$ on the group $\mathbf{R}_+^*$ does not vanish. (Identify $\mathscr{F}(f)$ with the function $y\mapsto \Gamma (1 +iy)$ and use FVR, VII.)

(h) We have

This statement is the “tauberian” theorem of Littlewood. The assumption concerning the sequence $(a_n)$, that is, $a_n= O(1/n)$, can moreover be weakened: it is sufficient that there exists a constant $A\geqslant 0$ such that $na_n\geqslant -A$ for every integer $n\geqslant 1$ (“Hardy–Littlewood tauberian theorem”, see for example J. Korevaar, Tauberian theory, Grundlehren math. wiss. 329, Springer (2004), th. 7.4, p. 16; the reader will take care not to confuse this theorem with that of FVR, I, p. 50, exercise 18).