Lemma 7

Let $f\in \mathscr{D}(\mathbf{R})$ and let $\widetilde{f}\in \mathscr{D}(\mathbf{C})$ be an almost analytic extension of $f$. For $\varepsilon >0$, define $f_{\varepsilon}$ on $\mathbf{R}$ by

Then $f_{\varepsilon}$ is continuous and bounded, and $f_{\varepsilon}$ converges to $f$ in $\mathscr{C}_b(\mathbf{R})$ when $\varepsilon$ tends to 0.

The continuity of $f_{\varepsilon}$ is a consequence of INT, IV, p. 144, § 4, n$^o3$, cor. 1, since $\widetilde{f}$ has compact support. Moreover, if $r$ is such that the support of $\widetilde{f}$ is contained in $[-r, r]\times \mathbf{R}$, one has

therefore $f_{\varepsilon}$ is bounded.

The Taylor expansion to order 1 (FVR, I, p. 30, prop. 3) and formula (13) prove that there exist $M\geqslant 0$ and a function $\varrho_1$ on $\mathbf{R}^2$ such that

$\widetilde{f}(y+i\gamma ) =f(y) +i\gamma f'(y) +\gamma^2\varrho_1(y;\gamma )$, and $|\varrho_1(y;\gamma )|\leqslant M$,

for every $y\in \mathbf{R}$ and $\gamma \in \mathbf{R}$. Since the mapping $y\mapsto \widetilde{f}(y+i\gamma )$ has support contained in $[-r, r]$ for every $\gamma \in \mathbf{R}$, the mapping $y\mapsto \varrho_1(y;\gamma )$ has support contained in $[-r, r]$, for every $\gamma \in \mathbf{R}$.

Let $g_{\varepsilon}$ be the function defined on $\mathbf{R}^2$ by

For every $(x, y)\in \mathbf{R}^2$ and $\varepsilon >0$, one obtains

Let $x\in \mathbf{R}$ and $\varepsilon >0$. We have

Consequently, for every $x\in \mathbf{R}$, there follows

where $\delta_{\varepsilon}$ and $\sigma_{\varepsilon}$ are the functions on $\mathbf{R}$ defined by

and $\|k_{\varepsilon}\|_{\infty}\leqslant 2Mr\varepsilon$.

The function $\mathscr{F}(f')\in \mathscr{S}(\mathbf{R})$ is integrable (Prop. 18 of IV, p. 219 and Prop. 13 of IV, p. 213). By Lemma 6, there follows

therefore $\mathscr{F}(f')\mathscr{F}(\sigma_{\varepsilon})$ converges to 0 in $L^1(\mathbf{R})$ when $\varepsilon$ tends to 0. Since $f'$ and $\sigma_{\varepsilon}$ belong to $L^2(\mathbf{R})$, Prop. 14 of II, p. 223 implies that $f'*\sigma_{\varepsilon}=\overline{\mathscr{F}}(\mathscr{F}(f')\mathscr{F}(\sigma_{\varepsilon}))$ converges to 0 in $\mathscr{C}_b(\mathbf{R})$.

For every $\varepsilon >0$, the positive measure $\delta_{\varepsilon}\cdot dx$ on $\mathbf{R}$ has total mass 1 (cf. FVR, III, p. 7). The set of measures $\delta_{\varepsilon}\cdot dx$ for $\varepsilon >0$ and the filter induced on this set by the filter of neighbourhoods of 0 in $\mathbf{R}_+^*$ satisfy the hypotheses of Lemma 4 of INT, VIII, p. 137, § 2, n$^o7$. The measures $\delta_{\varepsilon}\cdot dx$ therefore converge in $\mathscr{M}^1(\mathbf{R})$ to the point measure $\varepsilon_0$ when $\varepsilon$ tends to 0. There follows $f*\delta_{\varepsilon}\rightarrow f$ in $\mathscr{C}_b(\mathbf{R})$ (INT, VIII, p. 163, § 4, n$^o4$). The lemma is proved.

Let $\mu$ denote the Lebesgue measure on $\mathbf{C}$.