grandes-ecoles 2012 QI.A.1

grandes-ecoles · France · centrale-maths1__mp Sequences and Series Proof of Inequalities Involving Series or Sequence Terms

In each of the two cases below, show that $f * g$ is defined and bounded on $\mathbb{R}$ and give an upper bound for $\|f * g\|_{\infty}$ which may involve $\|\cdot\|_{1}$, $\|\cdot\|_{2}$ or $\|\cdot\|_{\infty}$. a) $f \in L^{1}(\mathbb{R}),\ g \in C_{b}(\mathbb{R})$; b) $f, g \in L^{2}(\mathbb{R})$.

In each of the two cases below, show that $f * g$ is defined and bounded on $\mathbb{R}$ and give an upper bound for $\|f * g\|_{\infty}$ which may involve $\|\cdot\|_{1}$, $\|\cdot\|_{2}$ or $\|\cdot\|_{\infty}$.\\
a) $f \in L^{1}(\mathbb{R}),\ g \in C_{b}(\mathbb{R})$;\\
b) $f, g \in L^{2}(\mathbb{R})$.

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