grandes-ecoles 2012 QIII.A.1

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

To any function $g$ in $C(\mathbb{R})$, we associate the linear form $\varphi_{g}$ on $L^{1}(\mathbb{R})$ defined by $$\varphi_{g}(f) = \int_{\mathbb{R}} f(t) g(-t) \mathrm{d}t$$ Let $(g_{1}, \ldots, g_{p})$ be a family of elements of $C_{b}(\mathbb{R})$. Show that the family $(g_{1}, \ldots, g_{p})$ is free if and only if the family $(\varphi_{g_{1}}, \ldots, \varphi_{g_{p}})$ is free.

To any function $g$ in $C(\mathbb{R})$, we associate the linear form $\varphi_{g}$ on $L^{1}(\mathbb{R})$ defined by
$$\varphi_{g}(f) = \int_{\mathbb{R}} f(t) g(-t) \mathrm{d}t$$
Let $(g_{1}, \ldots, g_{p})$ be a family of elements of $C_{b}(\mathbb{R})$.\\
Show that the family $(g_{1}, \ldots, g_{p})$ is free if and only if the family $(\varphi_{g_{1}}, \ldots, \varphi_{g_{p}})$ is free.

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