grandes-ecoles 2012 QII.A

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

For any function $f \in L^{1}(\mathbb{R})$, the Fourier transform of $f$ is the function $\hat{f}$ defined by $$\forall x \in \mathbb{R} \quad \hat{f}(x) = \int_{\mathbb{R}} f(t) \mathrm{e}^{-\mathrm{i}xt} \mathrm{~d}t$$ For any function $f \in L^{1}(\mathbb{R})$, show that $\hat{f}$ belongs to $C_{b}(\mathbb{R})$.

For any function $f \in L^{1}(\mathbb{R})$, the Fourier transform of $f$ is the function $\hat{f}$ defined by
$$\forall x \in \mathbb{R} \quad \hat{f}(x) = \int_{\mathbb{R}} f(t) \mathrm{e}^{-\mathrm{i}xt} \mathrm{~d}t$$
For any function $f \in L^{1}(\mathbb{R})$, show that $\hat{f}$ belongs to $C_{b}(\mathbb{R})$.

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