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zA)vBG?E|b%aPdXJJN%BDbK&(wzzvju2)sWR5P@vE<6?d-BsK(hX&e7ek>Xx#!%U~M zuC6Y$t8^G4btZc-1W0$ und eine Ordnung der Singularität $s \in \R$ existieren, sodass - \begin{align*}\label{math:sem:glatt} % Glatter KERN + \begin{align}\label{math:sem:glatt} % Glatter KERN \abs{\partial_{\bs x}^{\alpha}\partial_{\bs y}^{\beta}\kappa(\bs x, \bs y)} &\leq c_1(c_2(\abs{\bs x - \bs y})^{-(\abs{\alpha}+\abs{\beta}+s)}(\abs{\alpha} + \abs{\beta})! - \end{align*} + \end{align} für alle Multiindizes $\alpha,\beta \in \N_0^d$ mit $\abs{\alpha}+\abs{\beta}\geq 1$ gilt. \end{defi} Wie wir im Folgenden sehen werden, lassen sich asymptotisch glatte Kernfunktionen besonders gut durch Polynome interpolieren. Hierzu benötigen wir das Lemma \todo{cite}: -\begin{lem}\label{math:sem:ipol} %Interpol über Glatten KERN 1D -Sei $J \subseteq \R$ ein abgeschlossenes und beschränktes Intervall und angenommen die Funktion $u$ erfüllt für Konstanten $C_u,\rho_u > 0$ -\begin{align*} - \norm{u^{(n)}}_{\infty,J} &\leq C_u \rho_u^nn!\text{~~~für alle} n \in \N_0. -\end{align*} -Dann gilt für alle $k\in \N_0$ -\begin{align*} -\min_{v \in \P_k} \norm{u-v}_{\infty,J} &\leq C_u 4e(1+\rho_u\abs{J})(k+1)\left(1+\frac{2}{\rho_u\abs{J}}\right)^{-(k+1)}. -\end{align*} -\end{lem} -\hfill$\square$ +% \begin{lem}\label{math:sem:ipol} %Interpol über Glatten KERN 1D +% Sei $J \subseteq \R$ ein abgeschlossenes und beschränktes Intervall und angenommen die Funktion $u$ erfüllt für Konstanten $C_u,\rho_u > 0$ +% \begin{align*} +% \norm{\partial^n u}_{\infty,J} &\leq C_u \rho_u^nn!\text{~~~für alle} n \in \N_0. +% \end{align*} +% Dann gilt für alle $k\in \N_0$ +% \begin{align*} +% \min_{v \in \P_k} \norm{u-v}_{\infty,J} &\leq C_u 4e(1+\rho_u\abs{J})(k+1)\left(1+\frac{2}{\rho_u\abs{J}}\right)^{-(k+1)}. +% \end{align*} +% \end{lem} +% \hfill$\square$ -\begin{lem}\label{math:sem:ipolnD} %Interpol über Glatten KERN nD +\begin{lem}\label{thm:sem:ipolnD} %Interpol über Glatten KERN nD Sei $B \subseteq \R^d$ ein abgeschlossenes und beschränktes Intervall und angenommen die Funktion $u$ erfüllt für Konstanten $C_u,\rho_u > 0$ -\begin{align*} - \norm{\partial_j^nu}_{\infty,B} &\leq C_u \rho_u^nn!\text{~~~für alle} n \in \N_0. -\end{align*} +\begin{align} + \norm{\partial_j^nu}_{\infty,B} &\leq C_u \rho_u^nn!\quad\text{~für alle~} j \in \{1,\ldots,d\} \text{~und~} n \in \N_0. +\end{align} Dann gilt für alle $k\in \N_0$ -\begin{align*} -\min_{v \in \P_k} \norm{u-\I_k^Bu}_{\infty,B} &\leq C_u 8e\Lambda_k^d(1+\rho_u\diam (B))(k+1)\left(1+\frac{2}{\rho_u\diam(B)}\right)^{-(k+1)}. -\end{align*} +\begin{align}\label{math:sem:ipolnD} + \norm{u-\I_k^Bu}_{\infty,B} &\leq C_u 8e(1+\rho_u\diam (B))\Lambda_k^d(k+1)\left(1+\frac{2}{\rho_u\diam(B)}\right)^{-(k+1)}. +\end{align} \end{lem} \hfill$\square$ +\subsection{volle Quadratur} +\begin{defi} Seien $T_j,T_k$ Rechtecke und sei $\zeta_Q > 0$ fest. Dann heißt das Paar $(T_j,T_k)$ $\zeta_Q$-zulässig, genau dann wenn + \begin{align}\label{math:sem:zetaQ} + \dist (T_j, T_k)&\geq \zeta_Q \max\{ \diam(T_j) , \diam(T_k)\}. + \end{align} + Andernfalls heißt das Paar $(T_j,T_k)$ $\zeta_Q$-unzulässig. +\end{defi} -\begin{align*} - \Abs{\partial_x^n\partial_y^m\kappa}&=\Abs{\partial_x^n\partial_y^m\kappa(\gamma_j(x),\gamma_k(y))}\\ - &=\diam(T_j)^n \diam(T_k)^m \Abs{(\partial_x^n\partial_y^m\kappa)(\gamma_j(x),\gamma_k(y))} -\end{align*} - -\begin{align*} - \diam(T_j)^n& \diam(T_k)^m \Abs{(\partial_x^n\partial_y^m\kappa)(\gamma_j(x),\gamma_k(y))}\\ - \leq& \diam(T_j)^n \diam(T_k)^m c_1 (c_2 \abs{\gamma_j(x) - \gamma_k(y)})^{-(\abs{n}+\abs{m}+s)}(\abs{n}+\abs{m})!\\ - \leq & \max(\diam(T_j),\diam(T_k))^{n+m} c_1 (c_2 \abs{\bs x - \bs y})^{-(n+m+s)}(n+m)!\\ - = & \frac{c_1}{c_2 \abs{(\bs x - \bs y})^s} \left( \frac{\max(\diam(T_j),\diam(T_k))}{c_2 \abs{\bs x - \bs y}} \right)^{m+n}(\abs{n}+\abs{m})!\\ - \leq& C_{\kappa}\rho_{\kappa}^nn! -\end{align*} - -\begin{align*} - \min_{v \in \P_k} \norm{u-\I_k^Bu}_{\infty,B} &\leq C_k 8e\Lambda_k^d(1+\rho_u\diam (B))(k+1)\left(1+\frac{2}{\rho_u\diam(B)}\right)^{-(k+1)}. -\end{align*} - -\subsection{1D Integral} -\begin{lem} - Seien $T_j,T_k$ zwei $\zeta_1$-zulässige, affine Randstücke mit $\diam(T_j) \leq \diam(T_k)$. Weiterhin sei $\kappa$ eine asymptotisch glatte Kernfunktion mit den Konstanten $c_1,c_2>0$ und Singularitätsordung $s\geq0$. - Dann gilt mit - \begin{align} - C_{\zeta,j,\kappa} &:= 4 e\frac{c_1}{(c_2 \todo{\dist(T_j,T_k)})^s}\left(1+\frac{\zeta}{c_2}\right) - \end{align} - die Abschätzung +\begin{sat} Seien $T_j,T_k$ zwei $\zeta_Q$-zulässige Rechtecke. Sei $\kappa$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2>0$ und Singularitätsordung $s\geq0$. Dann gilt mit \begin{align*} - \sup_{\bs y \in T_k,x_b} \norm{\kappa(\cdot,\bs y)-\I_p\kappa(\cdot,\bs y)}_{\infty,T_{j| x_{\bs b}}} - &\leq C_{\zeta,j,\kappa}(1+\Lambda_p)(p+1)\left( 1+ \frac{2c_1}{\zeta} \right)^{-(p+1)}. + C_{\zeta_Q,j,k}&:=8e\frac{c_1}{(c_2\dist(T_j,T_k))^s} \left(1+\frac{\zeta_Q}{c_2} \right) + \end{align*} +die Abschätzung + \begin{align*} + \norm{\kappa(\cdot,\cdot)-\I_k^{T_j\times T_k}\kappa(\cdot,\cdot)}_{\infty,T_j\times T_k} + &\leq C_{\zeta,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-(p+1)}. + \end{align*} +\end{sat} +\beweis Zunächst definieren wir die Konstanten +\begin{align*} + C_{\kappa} &=\frac{c_1}{(c_2\dist(T_j,T_k))^s} \quad \text{und} \quad \rho_{\kappa} = \frac{\zeta_Q}{c_2} \end{align*} -\end{lem} -\beweis Wir definieren die Konstanten -\begin{align} - C_{\kappa} &:= \frac{c_1}{(c_2 \todo{\dist(T_j,T_k)})^s}\geq 0 \quad \text{und} \quad - \rho_{\kappa} := \frac{\zeta}{c_2}\geq 0 -\end{align} -und können die Konstante $C_{\zeta,j,\kappa}$ dann schreiben als: -\begin{align} - C_{\zeta,j,\kappa} &= C_{\kappa} 4 e(1+\rho_{\kappa}). -\end{align} -Sei $\bs y \in T_k$ fest gewählt. Ist $\gamma_{j | x_{\bs b}}$ die Parametrisierung von $T_{j| x_{\bs b}}$, so gilt aufgrund $\abs{\gamma_{j|x_{\bs b}}'}=\abs{a_j\bs a_j}=\diam_{|\bs a}(T_j)$ und der Kettenregel +und können dann die Konstante $C_{\zeta_Q,j,k}$ kurz \begin{align*} -\abs{\partial_{\bs x}^{\alpha}\kappa} -&= \abs{\partial_{x_{\bs a}}^n(\kappa(\gamma_{j | x_{\bs b}}(x_{\bs a}),\bs y))} -= (\diam_{|\bs a}(T_j))^n\abs{(\partial_{x_{\bs a}}^n\kappa)(\gamma_{j | x_{\bs b}}(x_{\bs a}),\bs y)}. + C_{\zeta_Q,j,k} &= 8eC_{\kappa} (1+\rho_{\kappa}) \end{align*} -Mithilfe von \eqref{math:sem:glatt} und den Konstanten $C_{\kappa},\rho_k$ gilt die Abschätzung\\ -\todo{Ist $\abs{\gamma_{j | x_{\bs b}}(x_{\bs a})-\bs y}$ wirklich $\abs{\bs x -\bs y} \Rightarrow \dist(T_j,T_k)$} +schreiben.\\ +Bezeichnet $\gamma$ die Parametrisierung von $T$, so gilt aufgrund der Kettenregel \begin{align*} - (\diam_{|\bs a}(T_j))^n&\abs{(\partial_{x_{\bs a}}^n\kappa)(\gamma_{j | x_{\bs b}}(x_{\bs a}),\bs y)}\\ - &\leq \diam_{|\bs a}(T_j))^n c_1(c_2 \abs{\gamma_{j | x_{\bs b}}(x_{\bs a})-\bs y})^{-n-s}n!\\ - &= \diam_{|\bs a}(T_j))^n c_1(c_2 \todo{\abs{\bs x-\bs y}})^{-n-s}n!\\ - &= \frac{c_1}{(c_2 \abs{\bs x-\bs y})^s} \left(\frac{\diam_{|\bs a}(T_j))}{c_2 \abs{\bs x-\bs y}} \right)^nn!\\ - &\leq C_{\kappa}\rho_{\kappa}^nn! + \Abs{\partial_x^{\alpha}\partial_y^{\beta}\kappa}&=\Abs{\partial_x^{\alpha}\partial_y^{\beta}\kappa(\gamma_j(x),\gamma_k(y))}\\ + &=\diam(T_j)^{\alpha} \diam(T_k)^{\beta} \Abs{(\partial_x^{\alpha}\partial_y^{\beta}\kappa)(\gamma_j(x),\gamma_k(y))}. \end{align*} -Da $T_j$ und $T_k$ zulässig sind ist $\kappa$ auf $T_j \times T_k$ glatt. Daher sind die Voraussetzungen für \eqref{math:sem:ipol} erfüllt und es gilt +Ferner gilt mit \eqref{math:sem:glatt} und den Konstanten $C_{\kappa},\rho_{\kappa}$ die Abschätzung \begin{align*} - \min_{v \in \P^p} \norm{\kappa(\gamma_{j | x_{\bs b}}(\cdot),\bs y)-v}_{\infty,[0,1]} - &\leq C_{\kappa} 4 e(1+\rho_{\kappa})(p+1)\left( 1+ \frac{2}{\rho_{\kappa}} \right)^{-(p+1)}\\ - &= C_{\zeta,j,\kappa}(p+1)\left( 1+ \frac{2}{\rho_{\kappa}} \right)^{-(p+1)} + \diam(T_j)^{\alpha}& \diam(T_k)^{\beta} \Abs{(\partial_x^{\alpha}\partial_y^{\beta}\kappa)(\gamma_j(x),\gamma_k(y))}\\ + \leq& \diam(T_j)^{\alpha} \diam(T_k)^{\beta} c_1 (c_2 \abs{\gamma_j(x) - \gamma_k(y)})^{-(\abs{\alpha}+\abs{\beta}+s)}(\abs{\alpha}+\abs{\beta})!\\ + \leq & \max(\diam(T_j),\diam(T_k))^{\alpha+\beta} c_1 (c_2 \abs{\bs x - \bs y})^{-(\alpha+\beta+s)}(\alpha+\beta)!\\ + = & \frac{c_1}{(c_2 \abs{\bs x - \bs y})^s} \left( \frac{\max(\diam(T_j),\diam(T_k))}{c_2 \abs{\bs x - \bs y}} \right)^{\alpha+\beta}(\abs{\alpha}+\abs{\beta})!\\ + \leq& C_{\kappa}\rho_{\kappa}^nn!. \end{align*} -für alle $p \in \N_0$, wobei $\P^p$ die Menge aller Polynome vom Grad $p$ ist. Daraus folgt also +Laut Voraussetzung sind $T_j,T_k$ $\zeta_Q$-zulässig, weshalb $\kappa$ auf $T_j,T_k$ definitionsgemäß glatt ist. Daher sind die Voraussetzungen für Lemma \ref{thm:sem:ipolnD} erfüllt und es gilt die Abschätzung mithilfe von \eqref{math:sem:zetaQ} \begin{align*} - \min_{v \in \P^p} \norm{\kappa(\gamma_{j | x_{\bs b}}(\cdot),\bs y)-v}_{\infty,[0,1]} - &\leq C_{\zeta,j,\kappa}(p+1)\left( 1+ \frac{2c_1}{\zeta} \right)^{-(p+1)} + \norm{\kappa(\gamma_j(\cdot),\gamma_k(\cdot))&-\I_p^{[0,1]^4}\kappa(\gamma_j(\cdot),\gamma_k(\cdot))}_{\infty,[0,1]^4}\\ + &\leq C_{\kappa} 8e(1+\rho_{\kappa})\Lambda_k^4(p+1)\left(1+\frac{2}{\rho_{\kappa}}\right)^{-(p+1)}\\ +% &= C_{\zeta,j,k}\Lambda_k^4(p+1)\left(1+\frac{2}{\rho_{\kappa}}\right)^{-(p+1)}\\ + &= C_{\zeta,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-(p+1)}. \end{align*} -und mithilfe von Satz \ref{math:ipol} gilt +Weiterhin folgt für $T_j \times T_k$ \begin{align*} - \norm{\kappa(\cdot,\bs y)-\I_p\kappa(\cdot,\bs y)}_{\infty,T_{j| x_{\bs b}}} - &\leq C_{\zeta,j,\kappa}(1+\Lambda_p)(p+1)\left( 1+ \frac{2c_1}{\zeta} \right)^{-(p+1)}. + \norm{\kappa(\cdot,\cdot)-\I_k^{T_j\times T_k}\kappa(\cdot,\cdot)}_{\infty,T_j\times T_k} + &\leq C_{\zeta,j,k}\Lambda_k^4(p+1)\left(1+\frac{2c_2}{\zeta_Q}\right)^{-(p+1)}, \end{align*} -Da $\bs y$ und $x_{\bs b}$ beliebig war und die rechte Seite unabhängig von $\bs y,x_{\bs b}$ ist, folgt die Behauptung für das Supremum. +womit der Beweis abgeschlossen ist. \hfill$\square$ +% \subsection{1D Integral} +% \begin{lem} +% Seien $T_j,T_k$ zwei $\zeta_1$-zulässige, affine Randstücke mit $\diam(T_j) \leq \diam(T_k)$. Weiterhin sei $\kappa$ eine asymptotisch glatte Kernfunktion mit den Konstanten $c_1,c_2>0$ und Singularitätsordung $s\geq0$. +% Dann gilt mit +% \begin{align} +% C_{\zeta,j,\kappa} &:= 4 e\frac{c_1}{(c_2 \todo{\dist(T_j,T_k)})^s}\left(1+\frac{\zeta}{c_2}\right) +% \end{align} +% die Abschätzung +% \begin{align*} +% \sup_{\bs y \in T_k,x_b} \norm{\kappa(\cdot,\bs y)-\I_p\kappa(\cdot,\bs y)}_{\infty,T_{j| x_{\bs b}}} +% &\leq C_{\zeta,j,\kappa}(1+\Lambda_p)(p+1)\left( 1+ \frac{2c_1}{\zeta} \right)^{-(p+1)}. +% \end{align*} +% \end{lem} +% \beweis Wir definieren die Konstanten +% \begin{align} +% C_{\kappa} &:= \frac{c_1}{(c_2 \todo{\dist(T_j,T_k)})^s}\geq 0 \quad \text{und} \quad +% \rho_{\kappa} := \frac{\zeta}{c_2}\geq 0 +% \end{align} +% und können die Konstante $C_{\zeta,j,\kappa}$ dann schreiben als: +% \begin{align} +% C_{\zeta,j,\kappa} &= C_{\kappa} 4 e(1+\rho_{\kappa}). +% \end{align} +% Sei $\bs y \in T_k$ fest gewählt. Ist $\gamma_{j | x_{\bs b}}$ die Parametrisierung von $T_{j| x_{\bs b}}$, so gilt aufgrund $\abs{\gamma_{j|x_{\bs b}}'}=\abs{a_j\bs a_j}=\diam_{|\bs a}(T_j)$ und der Kettenregel +% \begin{align*} +% \abs{\partial_{\bs x}^{\alpha}\kappa} +% &= \abs{\partial_{x_{\bs a}}^n(\kappa(\gamma_{j | x_{\bs b}}(x_{\bs a}),\bs y))} +% = (\diam_{|\bs a}(T_j))^n\abs{(\partial_{x_{\bs a}}^n\kappa)(\gamma_{j | x_{\bs b}}(x_{\bs a}),\bs y)}. +% \end{align*} +% Mithilfe von \eqref{math:sem:glatt} und den Konstanten $C_{\kappa},\rho_k$ gilt die Abschätzung\\ +% \todo{Ist $\abs{\gamma_{j | x_{\bs b}}(x_{\bs a})-\bs y}$ wirklich $\abs{\bs x -\bs y} \Rightarrow \dist(T_j,T_k)$} +% \begin{align*} +% (\diam_{|\bs a}(T_j))^n&\abs{(\partial_{x_{\bs a}}^n\kappa)(\gamma_{j | x_{\bs b}}(x_{\bs a}),\bs y)}\\ +% &\leq \diam_{|\bs a}(T_j))^n c_1(c_2 \abs{\gamma_{j | x_{\bs b}}(x_{\bs a})-\bs y})^{-n-s}n!\\ +% &= \diam_{|\bs a}(T_j))^n c_1(c_2 \todo{\abs{\bs x-\bs y}})^{-n-s}n!\\ +% &= \frac{c_1}{(c_2 \abs{\bs x-\bs y})^s} \left(\frac{\diam_{|\bs a}(T_j))}{c_2 \abs{\bs x-\bs y}} \right)^nn!\\ +% &\leq C_{\kappa}\rho_{\kappa}^nn! +% \end{align*} +% Da $T_j$ und $T_k$ zulässig sind ist $\kappa$ auf $T_j \times T_k$ glatt. Daher sind die Voraussetzungen für \eqref{math:sem:ipolnD} erfüllt und es gilt +% \begin{align*} +% \min_{v \in \P^p} \norm{\kappa(\gamma_{j | x_{\bs b}}(\cdot),\bs y)-v}_{\infty,[0,1]} +% &\leq C_{\kappa} 4 e(1+\rho_{\kappa})(p+1)\left( 1+ \frac{2}{\rho_{\kappa}} \right)^{-(p+1)}\\ +% &= C_{\zeta,j,\kappa}(p+1)\left( 1+ \frac{2}{\rho_{\kappa}} \right)^{-(p+1)} +% \end{align*} +% für alle $p \in \N_0$, wobei $\P^p$ die Menge aller Polynome vom Grad $p$ ist. Daraus folgt also +% \begin{align*} +% \min_{v \in \P^p} \norm{\kappa(\gamma_{j | x_{\bs b}}(\cdot),\bs y)-v}_{\infty,[0,1]} +% &\leq C_{\zeta,j,\kappa}(p+1)\left( 1+ \frac{2c_1}{\zeta} \right)^{-(p+1)} +% \end{align*} +% und mithilfe von Satz \ref{math:ipol} gilt +% \begin{align*} +% \norm{\kappa(\cdot,\bs y)-\I_p\kappa(\cdot,\bs y)}_{\infty,T_{j| x_{\bs b}}} +% &\leq C_{\zeta,j,\kappa}(1+\Lambda_p)(p+1)\left( 1+ \frac{2c_1}{\zeta} \right)^{-(p+1)}. +% \end{align*} +% Da $\bs y$ und $x_{\bs b}$ beliebig war und die rechte Seite unabhängig von $\bs y,x_{\bs b}$ ist, folgt die Behauptung für das Supremum. +% \hfill$\square$ + \subsection{Matrix} \begin{sat} Seien $Tj,Tk$ zwei $\zeta_1$-zulässige affine Randstücke mit zugehörigen Parametrisierungen $\gamma_j,\gamma_k$ und $\diam(T_j) \leq \diam(T_k)$. Sei weiterhin $\kappa : \R^3\times\R^3\to \R : (\bs x, \bs y) \mapsto \kappa(\bs x,\bs y)$ eine asymptotisch glatte Kernfunktion mit Konstanten $c_1,c_2,s$. Sei -- 2.47.3