grandes-ecoles 2013 QI.A.1

grandes-ecoles · France · centrale-maths1__psi Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences

Justify the equality
$$\forall t \in \mathbb { R } \quad G _ { x } ( t ) = e ^ { i x \sin t } = \sum _ { n = - \infty } ^ { + \infty } \varphi _ { n } ( x ) e ^ { i n t }$$
What can be said about the convergence of the Fourier series of $G _ { x }$ ?

Justify the equality

$$\forall t \in \mathbb { R } \quad G _ { x } ( t ) = e ^ { i x \sin t } = \sum _ { n = - \infty } ^ { + \infty } \varphi _ { n } ( x ) e ^ { i n t }$$

What can be said about the convergence of the Fourier series of $G _ { x }$ ?

Paper Questions

QI.A.1 QI.A.2 QI.B QI.C QI.D QII.A QII.B QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.D QII.E.1 QII.E.2 QII.E.3 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIV.A QIV.B QIV.C.1 QIV.C.2 QIV.C.3 QIV.D.1 QIV.D.2