Exercise 27, § 2

The object is to prove the converse of the assertion in question b) of exercise 25. For this one will use the results and the notations of exercise 23 of V, p. 503 below. In particular, G denotes the group $\mathbf{R}^*_+$ endowed with the Haar measure $\mu=x^{-1}dx$, dual to the group $i\mathbf{R}$ by the mapping $(x, it)\mapsto x^{it}$. Denote by X the identity mapping of G.

Denote by co (resp. si, resp. ta) the function $it\mapsto$ cos($i\pi t$) (resp. the function $it\mapsto$ sin($i\pi t$), the function $it\mapsto$ tan($i\pi t$)) on $i\mathbf{R}$.

Let $f\in$ Loe(G) satisfy conditions (i), (ii) and (iii) of question f ) of exercise 25.

a) Let $\alpha$ be the function $x\mapsto x/(1 +x)$ on G. One has $I_{\alpha}= ]-1,0[$ and $\widehat{\alpha}(s) =$ $-\pi /$ sin($\pi s$) for $s\in B_f$.

b) The function ta is bounded on $i\mathbf{R}$. Denote by $v=\mathscr{F}\circ m_{ta}\circ \overline{\mathscr{F}}$; this is an endomorphism of $L^2(G)$ such that $\|v\|\leqslant 1$ and $\mathscr{R}(v) = 0$.

c) Let $\varepsilon >0$ be a real number. Denote by $u_{\varepsilon}$ the endomorphism of multiplication by $h_{\varepsilon}= X/(1 +\varepsilon X)$ on $L^2(G)$. One has $u_{\varepsilon}\geqslant 0$ and for every $\varphi \in L^2(G)$ and every bounded continuous function $f$ on G, one has

1

2

where $Q(\varphi ) =\varphi v(\varphi ) +\overline{v(\varphi)}\varphi$.

d) Let F be the vector subspace of $L^2(i\mathbf{R})$ generated by the functions $t\mapsto e^{\alpha(it-s_0)^2}$ where $\alpha \in \mathbf{R}_+^*$ and $s_0\in \mathbf{C}$; it is dense in $L^2(i\mathbf{R})$. e) Let $\psi \in F$. There exists $\varphi \in L^2(G)$ such that $\widehat{\varphi}=$ co $\cdot \psi$. Let $g= Q(\varphi )$. One has $g\in L^1$(G), and there exists a holomorphic function on $\mathbf{C}$ whose restriction to $i\mathbf{R}$ coincides with $\widehat{g}$. Moreover

$\widehat{g}=\frac{1}{2i\pi}(\psi^**\psi )$ si.

f) For every $\varepsilon >0$, one has $\mathscr{R}(u_{\varepsilon}\circ v)\geqslant 0$. (Calculate $\langle \varphi |\mathscr{R}(u_{\varepsilon}u)\varphi \rangle$ when $\varphi$ is a function as in the preceding question, and use the convolution on G as in question d) of exercise 23 of V, p. 503.)

g) Let $f\in$ Loe($\mathbf{R}^*_+$). Let $g$ be a function associated with $\psi \in F$ as in question e). One then has

and

$\int_{\mathbf{R}}\widehat{f}(-\sigma -it)$ sin($\pi (\sigma +it)$)$(\psi^**\psi )(\sigma +it)dt\geqslant 0$

for every $\sigma \in ]0,1[$.

h) Let $\sigma \in ]0,1[$. The function $q$ defined by $q(s) =\widehat{f}(-s)$ sin($\pi s$) for $s$ such that $\mathscr{R}(s) =\sigma$ satisfies

$nn$ $\sum\sum\overline{t}_it_jq(\frac{1}{2}(\overline{z}_i+z_j))\geqslant 0$ $i=0j=0$

for every $n\in \mathbf{N}$ and every families $(t_i)_{0\leqslant i\leqslant n}$ and $(z_i)_{0\leqslant i\leqslant n}$ of complex numbers such that $\mathscr{R}(z_i) =\sigma$ for every $i$.

i) There exists a positive measure $\nu_0$ on G such that $\widehat{\nu}_0(it) =\widehat{f}(-it)$ sin($i\pi t$)$/\pi$ for every $t\in \mathbf{R}$; one has $X^{\sigma}\in L^1(G, \nu_0)$ if $0< \sigma <1$.

j) Let $\nu =\check{\nu}^0$. The measure $\nu$ is bounded. (Verify that$]-1,0[\subset I_{\nu}$ and then that $\widehat{\nu}(-t)$ is bounded when $t\rightarrow 0$ in order to prove that $\nu ([1,+\infty [)$ is finite.) For every $x$, one has