grandes-ecoles 2012 QI.D.2

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

Let $f \in C_{b}(\mathbb{R})$ and let $(\delta_{n})$ be a sequence of functions forming an approximate identity. Show that if $f$ has compact support, then the sequence $\left(f * \delta_{n}\right)_{n \in \mathbb{N}}$ converges uniformly to $f$ on $\mathbb{R}$.

Let $f \in C_{b}(\mathbb{R})$ and let $(\delta_{n})$ be a sequence of functions forming an approximate identity.\\
Show that if $f$ has compact support, then the sequence $\left(f * \delta_{n}\right)_{n \in \mathbb{N}}$ converges uniformly to $f$ on $\mathbb{R}$.

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