grandes-ecoles 2016 QV.B

grandes-ecoles · France · centrale-maths2__psi Differentiating Transcendental Functions Regularity and smoothness of transcendental functions

Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. Let $h$ be the function defined on $\mathbb{R}$, which is 1-periodic and which equals $\mathcal{F}(f)$ on the interval $[-1/2, 1/2]$. Show that $h$ is of class $C^{\infty}$ on $\mathbb{R}$.

Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. Let $h$ be the function defined on $\mathbb{R}$, which is 1-periodic and which equals $\mathcal{F}(f)$ on the interval $[-1/2, 1/2]$. Show that $h$ is of class $C^{\infty}$ on $\mathbb{R}$.

Paper Questions

QI.A QI.B QI.C QI.D QI.E QII.A QII.B QII.C QII.D QII.E QIII.A QIII.B QIII.C QIV.A QIV.B QIV.C QIV.D QIV.E QIV.F QIV.G QV.A QV.B QV.C QV.D QV.E QVI.A QVI.B QVI.C