grandes-ecoles 2012 QII.D.2

grandes-ecoles · France · centrale-maths1__mp Sequences and Series Functional Equations and Identities via Series

Let $f \in L^{1}(\mathbb{R})$ be such that $\hat{f} \in L^{1}(\mathbb{R})$. For every real $t$ and every non-zero natural number $n$, we set $$I_{n}(t) = \frac{1}{2\pi} \int_{\mathbb{R}} k_{n}(x) \hat{f}(-x) \mathrm{e}^{-\mathrm{i}tx} \mathrm{~d}x$$ Deduce, for every real $t$: $$f(t) = \frac{1}{2\pi} \int_{\mathbb{R}} \hat{f}(x) \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}x$$

Let $f \in L^{1}(\mathbb{R})$ be such that $\hat{f} \in L^{1}(\mathbb{R})$. For every real $t$ and every non-zero natural number $n$, we set
$$I_{n}(t) = \frac{1}{2\pi} \int_{\mathbb{R}} k_{n}(x) \hat{f}(-x) \mathrm{e}^{-\mathrm{i}tx} \mathrm{~d}x$$
Deduce, for every real $t$:
$$f(t) = \frac{1}{2\pi} \int_{\mathbb{R}} \hat{f}(x) \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}x$$

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