Fourier series. Let $\phi : \mathbb { R } \rightarrow \mathbb { C }$ be a periodic function with period $2 \pi$, of class $\mathcal { C } ^ { 1 }$. (a) Show that for all $n \in \mathbb { Z } ^ { * } , c _ { n } ( \phi ) = \frac { c _ { n } \left( \phi ^ { \prime } \right) } { \text { in } }$. (b) Show that the series $\sum _ { n \in \mathbb { Z } } \left| c _ { n } ( \phi ) \right|$ converges. Hint. Use Parseval's formula for the function $\phi ^ { \prime }$. (c) Show that $\| \phi \| _ { \infty } \leq \sum _ { n \in \mathbb { Z } } \left| c _ { n } ( \phi ) \right|$.
Fourier series. Let $\phi : \mathbb { R } \rightarrow \mathbb { C }$ be a periodic function with period $2 \pi$, of class $\mathcal { C } ^ { 1 }$.\\
(a) Show that for all $n \in \mathbb { Z } ^ { * } , c _ { n } ( \phi ) = \frac { c _ { n } \left( \phi ^ { \prime } \right) } { \text { in } }$.\\
(b) Show that the series $\sum _ { n \in \mathbb { Z } } \left| c _ { n } ( \phi ) \right|$ converges. Hint. Use Parseval's formula for the function $\phi ^ { \prime }$.\\
(c) Show that $\| \phi \| _ { \infty } \leq \sum _ { n \in \mathbb { Z } } \left| c _ { n } ( \phi ) \right|$.