grandes-ecoles 2012 QII.B.1

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

Let $f, g \in L^{1}(\mathbb{R})$. Assume that $g$ is bounded. a) Show that $f * g$ is integrable on $\mathbb{R}$ and determine $\int_{\mathbb{R}} f * g$ in terms of $\int_{\mathbb{R}} f$ and $\int_{\mathbb{R}} g$. b) Show that $\widehat{f * g} = \hat{f} \times \hat{g}$.

Let $f, g \in L^{1}(\mathbb{R})$. Assume that $g$ is bounded.\\
a) Show that $f * g$ is integrable on $\mathbb{R}$ and determine $\int_{\mathbb{R}} f * g$ in terms of $\int_{\mathbb{R}} f$ and $\int_{\mathbb{R}} g$.\\
b) Show that $\widehat{f * g} = \hat{f} \times \hat{g}$.

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