Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a $2 \pi$-periodic function of class $\mathcal { C } ^ { 1 }$. We consider the Fourier series of $f$ in cosines and sines, denoted $$c _ { 0 } + \sum _ { n \geq 1 } \left( a _ { n } \cos ( n t ) + b _ { n } \sin ( n t ) \right)$$ Show that, for all $x \in ] - 1,1 [$ and all $t \in \mathbb { R }$, $$c _ { 0 } + \sum _ { n = 1 } ^ { + \infty } \left( a _ { n } \cos ( n t ) + b _ { n } \sin ( n t ) \right) x ^ { n } = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \frac { \left( 1 - x ^ { 2 } \right) f ( u ) } { x ^ { 2 } - 2 x \cos ( t - u ) + 1 } \mathrm { ~d} u$$
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a $2 \pi$-periodic function of class $\mathcal { C } ^ { 1 }$. We consider the Fourier series of $f$ in cosines and sines, denoted
$$c _ { 0 } + \sum _ { n \geq 1 } \left( a _ { n } \cos ( n t ) + b _ { n } \sin ( n t ) \right)$$
Show that, for all $x \in ] - 1,1 [$ and all $t \in \mathbb { R }$,
$$c _ { 0 } + \sum _ { n = 1 } ^ { + \infty } \left( a _ { n } \cos ( n t ) + b _ { n } \sin ( n t ) \right) x ^ { n } = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \frac { \left( 1 - x ^ { 2 } \right) f ( u ) } { x ^ { 2 } - 2 x \cos ( t - u ) + 1 } \mathrm { ~d} u$$