grandes-ecoles 2012 QI.C.3

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

In this question I.C.3, assume that $g$ is continuous, $2\pi$-periodic and of class $C^{1}$ piecewise. a) State without proof the theorem on Fourier series applicable to continuous, $2\pi$-periodic functions of class $C^{1}$ piecewise. b) Show that $f * g$ is $2\pi$-periodic and is the sum of its Fourier series. Specify the Fourier coefficients of $f * g$ using the Fourier coefficients of $g$ and integrals involving $f$.

In this question I.C.3, assume that $g$ is continuous, $2\pi$-periodic and of class $C^{1}$ piecewise.\\
a) State without proof the theorem on Fourier series applicable to continuous, $2\pi$-periodic functions of class $C^{1}$ piecewise.\\
b) Show that $f * g$ is $2\pi$-periodic and is the sum of its Fourier series. Specify the Fourier coefficients of $f * g$ using the Fourier coefficients of $g$ and integrals involving $f$.

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