grandes-ecoles 2012 QIII.C.4

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 in $L^{1}(\mathbb{R})$. The functions $h_{r}$ are those from question I.D.3, defined on $[-1,1]$ by $h_{r}(t) = \frac{(1-t^2)^r}{\lambda_r}$ and zero outside $[-1,1]$. Show that for every non-zero natural number $r$, $V_{h_{r} * g}$ is finite-dimensional.

Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension in $L^{1}(\mathbb{R})$. The functions $h_{r}$ are those from question I.D.3, defined on $[-1,1]$ by $h_{r}(t) = \frac{(1-t^2)^r}{\lambda_r}$ and zero outside $[-1,1]$.\\
Show that for every non-zero natural number $r$, $V_{h_{r} * g}$ is finite-dimensional.

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