grandes-ecoles 2010 QIII.C.2

grandes-ecoles · France · centrale-maths1__pc Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences

For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$
Let $f \in C^\infty([-1,1])$.
Show that the Fourier series of $\widetilde{f}$ converges normally to $\widetilde{f}$.

For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function:
$$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$

Let $f \in C^\infty([-1,1])$.

Show that the Fourier series of $\widetilde{f}$ converges normally to $\widetilde{f}$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.B.1 QI.B.2 QI.B.3 QI.C.1 QI.C.2 QI.C.3 QI.C.4 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B.1 QII.B.2 QII.B.3 QII.C QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.D.1 QIII.D.2