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\displaystyle + \frac{1}{2\pi h} + \int + e^{-i\omega \frac{z-Z^*}{h}} + \cdot \frac - {\phi_{K_{h, \lVert \cdot \rVert}}^{[\, z\, ]}( \omega \, ; \, h) \phi_K(\omega)} - {\phi_V(\omega)} - \, d\omega + { + \phi_{ + K_{\, \lVert \, \bullet \, \rVert}^{[\,\int \,]} + } + (\omega \,, z\, \vert \, h) + \phi_K(\omega) + } + { + \phi_u\left(\frac \omega h\right) + } + d\omega \end{equation}\label{eq:norm_deconvolution_kernel} -où $\quad \phi_{K_{h, \lVert \cdot \rVert}}^{[\, z\, ]}( \omega \, ; \, h) = \int e^{i \omega \left(\frac{z-u}{h}\right)} K_{h, \lVert \cdot \rVert}^{[\, u \,]}(z) du$ +où $\phi_{K_{\, \lVert \, \bullet \, \rVert}^{[\,\int \,]}}(\omega \,, z\, \vert \, h) = \int e^{i \omega \left(\frac{z-u}{h}\right)} K_{h, \lVert \cdot \rVert}^{[\, u \,]}(z) \, du$ est la transformée de fourier selon les points de centrage (les $Z^*$ observés). \subsubsection{Propriétés asymptotiques} diff --git a/src/content/pre/notations/main.tex b/src/content/pre/notations/main.tex index 6be1ebb..e6107b0 100644 --- a/src/content/pre/notations/main.tex +++ b/src/content/pre/notations/main.tex @@ -61,6 +61,8 @@ \section*{Notations}\label{notations} $\phi_X$ & Fonction caractéristique de la variable aléatoire $X$ : \linebreak $$\phi_X : \func{\grandR}{\mathds C}{\omega}{\esperance{e^{i \omega X}}}$$ \\ $\phi_{\textsf{erreur}}$ & Fonction caractéristique de l'erreur de mesure $U$ aussi appelée $\phi_u$ quand il n'y a pas d'ambiguïté \\ + $\phi_{K_{\, \lVert \, \bullet \, \rVert}^{[\,\int \,]}}(\omega \,, x\, \vert \, h)$ & transformée de fourier du noyau de lissage normalisé selon les points de centrage au point $x$ pour la fréquence $\omega$ : $\int e^{i \omega \frac{x-\mathbf u}h} K_{h, \Vert \cdot \Vert}^{[\mathbf u]}(x) d\mathbf u$ + \\ \midrule \textbf{Noyaux} & @@ -86,7 +88,7 @@ \section*{Notations}\label{notations} & Noyau de déconvolution basé sur le noyau de lissage $K$ : \linebreak $$\tilde K_h : \func{\grandR}{\Rplus}{t}{\displaystyle\frac 1 {2\pi} \int e^{-iut} \frac{\mathcal F_{\textsf{stat}}[K](hu)}{\phi_{\textsf{erreur}}(u)} du} $$ \\\\ $\tilde K_h^\star$ - & Noyau de déconvolution normalisé basé sur le noyau de lissage $K$ :\linebreak $$\tilde K_h^\star : \func{\grandR}{\Rplus}{t}{\displaystyle\frac{1}{2\pi h}\int e^{-i\omega \frac{x-X^*}{h}} \cdot \frac{\phi_{K, \, \Vert \, \cdot \, \Vert}^{[\,x\,]}(\omega \,;\, h)\phi_K(\omega)}{\phi_u\left(\frac \omega h\right)}}$$ + & Noyau de déconvolution normalisé basé sur le noyau de lissage $K$ :\linebreak $$\tilde K_h^\star : \func{\grandR}{\Rplus}{x}{\displaystyle\frac{1}{2\pi h}\int e^{-i\omega \frac{x-X^*}{h}} \cdot \frac{ \phi_{K_{\, \lVert \, \bullet \, \rVert}^{[\,\int \,]}}(\omega \,, x\, \vert \, h) \phi_K(\omega)}{\phi_u\left(\frac \omega h\right)}d\omega} $$ \\ \bottomrule \end{tabularx}