grandes-ecoles 2012 QIII.C.3

grandes-ecoles · France · centrale-maths1__mp Sequences and Series Properties and Manipulation of Power Series or Formal Series

Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension $n$ in $L^{1}(\mathbb{R})$, and let $\alpha_{1}, \ldots, \alpha_{n}$, $m_{1}, \ldots, m_{n}$ be as in III.C.1 such that $T_{\alpha}(g) = \sum_{i=1}^{n} m_{i}(\alpha) T_{\alpha_{i}}(g)$ for every real $\alpha$. By applying question III.C.2) to $V_{g}$, show that if $g$ is of class $C^{k}$ then the functions $m_{1}, \ldots, m_{n}$ are of class $C^{k}$.

Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension $n$ in $L^{1}(\mathbb{R})$, and let $\alpha_{1}, \ldots, \alpha_{n}$, $m_{1}, \ldots, m_{n}$ be as in III.C.1 such that $T_{\alpha}(g) = \sum_{i=1}^{n} m_{i}(\alpha) T_{\alpha_{i}}(g)$ for every real $\alpha$.\\
By applying question III.C.2) to $V_{g}$, show that if $g$ is of class $C^{k}$ then the functions $m_{1}, \ldots, m_{n}$ are of class $C^{k}$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.B.1 QI.B.2 QI.B.3 QI.B.4 QI.B.5 QI.C.1 QI.C.2 QI.C.3 QI.D.1 QI.D.2 QI.D.3 QI.D.4 QII.A QII.B.1 QII.B.2 QII.C.1 QII.C.2 QII.C.3 QII.D.1 QII.D.2 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.B.1 QIII.B.2 QIII.C.1 QIII.C.2 QIII.C.3 QIII.C.4 QIII.C.5 QIII.C.6 QIII.C.7