grandes-ecoles 2012 QIII.C.1

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

Let $g \in C_{b}(\mathbb{R})$. We assume that $N_{g}$ has finite codimension $n$ in $L^{1}(\mathbb{R})$, and that $V_{g} = \operatorname{Vect}\left(T_{\alpha}(g)\right)_{\alpha \in \mathbb{R}}$ has dimension $n$. Show that there exist real numbers $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ and functions $m_{1}, \ldots, m_{n}$ of a real variable such that, for every real $\alpha$, $$T_{\alpha}(g) = \sum_{i=1}^{n} m_{i}(\alpha) T_{\alpha_{i}}(g)$$

Let $g \in C_{b}(\mathbb{R})$. We assume that $N_{g}$ has finite codimension $n$ in $L^{1}(\mathbb{R})$, and that $V_{g} = \operatorname{Vect}\left(T_{\alpha}(g)\right)_{\alpha \in \mathbb{R}}$ has dimension $n$.\\
Show that there exist real numbers $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ and functions $m_{1}, \ldots, m_{n}$ of a real variable such that, for every real $\alpha$,
$$T_{\alpha}(g) = \sum_{i=1}^{n} m_{i}(\alpha) T_{\alpha_{i}}(g)$$

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