grandes-ecoles 2012 QIII.B.1

grandes-ecoles · France · centrale-maths1__mp Second order differential equations Structure of the solution space

Let $g \in C_{b}(\mathbb{R})$. We say that $g$ satisfies hypothesis A if $g$ is a function of class $C^{\infty}$ on $\mathbb{R}$, bounded and whose derivative functions of all orders are bounded on $\mathbb{R}$. Show that if $N_{g}$ has finite codimension in $L^{1}(\mathbb{R})$ and if $g$ satisfies hypothesis A, then $g$ is a solution of a linear differential equation with constant coefficients.

Let $g \in C_{b}(\mathbb{R})$. We say that $g$ satisfies hypothesis A if $g$ is a function of class $C^{\infty}$ on $\mathbb{R}$, bounded and whose derivative functions of all orders are bounded on $\mathbb{R}$.\\
Show that if $N_{g}$ has finite codimension in $L^{1}(\mathbb{R})$ and if $g$ satisfies hypothesis A, then $g$ is a solution of a linear differential equation with constant coefficients.

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