grandes-ecoles 2016 QIII.C

grandes-ecoles · France · centrale-maths2__psi Taylor series Prove smoothness or power series expandability of a function

Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
We assume that $f$ is zero outside a segment $[a, b]$. Show that $f = 0$.

Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have

$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$

We assume that $f$ is zero outside a segment $[a, b]$. Show that $f = 0$.

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