grandes-ecoles 2012 QI.B.5

grandes-ecoles · France · centrale-maths1__mp Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences

For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. Show that $f * g$ is uniformly continuous on $\mathbb{R}$ in the general case.

For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$.\\
Show that $f * g$ is uniformly continuous on $\mathbb{R}$ in the general case.

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