grandes-ecoles 2015 QI.A.4

grandes-ecoles · France · centrale-maths2__pc Differentiating Transcendental Functions Regularity and smoothness of transcendental functions

For every function $f$ belonging to $\mathcal{F}_{sr}$ and every non-zero natural number $n$, we set $$\left(f * \rho_n\right)(x) = \int_{\mathbb{R}} f(t) \rho_n(x-t) \mathrm{d}t$$
Let $f$ be a function belonging to $\mathcal{F}_{sr}$. Show that the function $f * \rho_n$ is of class $\mathcal{C}^{\infty}$.

For every function $f$ belonging to $\mathcal{F}_{sr}$ and every non-zero natural number $n$, we set
$$\left(f * \rho_n\right)(x) = \int_{\mathbb{R}} f(t) \rho_n(x-t) \mathrm{d}t$$

Let $f$ be a function belonging to $\mathcal{F}_{sr}$. Show that the function $f * \rho_n$ is of class $\mathcal{C}^{\infty}$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.B.1 QI.B.2 QI.B.3 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QII.B.5 QII.C.1 QII.C.2