grandes-ecoles 2012 QIII.B.2

grandes-ecoles · France · centrale-maths1__mp Second order differential equations Qualitative and asymptotic analysis of solutions

Let $g \in C_{b}(\mathbb{R})$ satisfying hypothesis A (i.e., $g$ is of class $C^{\infty}$ on $\mathbb{R}$, bounded, and all its derivative functions of all orders are bounded on $\mathbb{R}$). Deduce the set of functions $g$ satisfying hypothesis A and such that $N_{g}$ has finite codimension in $L^{1}(\mathbb{R})$.

Let $g \in C_{b}(\mathbb{R})$ satisfying hypothesis A (i.e., $g$ is of class $C^{\infty}$ on $\mathbb{R}$, bounded, and all its derivative functions of all orders are bounded on $\mathbb{R}$).\\
Deduce the set of functions $g$ satisfying hypothesis A and such that $N_{g}$ has finite codimension in $L^{1}(\mathbb{R})$.

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