Let $\psi : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function, periodic with period $2 \pi$. Let $f : [ a , b ] \rightarrow \mathbb { R }$ be a function of class $\mathcal { C } ^ { 1 }$ on $[ a , b ]$. For every parameter $\varepsilon > 0$, we set $$J _ { \varepsilon } = \int _ { a } ^ { b } \psi \left( \frac { x } { \varepsilon } \right) f ( x ) d x$$ First case. In this question, we further assume that $\psi$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R }$ and that $f$ has compact support in $] a , b [$. (a) Show that for all $\varepsilon > 0$, $$\left| J _ { \varepsilon } - c _ { 0 } ( \psi ) \left( \int _ { a } ^ { b } f ( x ) d x \right) \right| \leq \varepsilon ( b - a ) \left\| f ^ { \prime } \right\| _ { \infty } \sum _ { n \in \mathbb { Z } ^ { * } } \frac { \left| c _ { n } ( \psi ) \right| } { | n | }$$ Hint. One can reduce to the case where $\int _ { 0 } ^ { 2 \pi } \psi ( y ) d y = 0$. (b) Deduce the limit of $J _ { \varepsilon }$ as $\varepsilon \rightarrow 0$.
Let $\psi : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function, periodic with period $2 \pi$. Let $f : [ a , b ] \rightarrow \mathbb { R }$ be a function of class $\mathcal { C } ^ { 1 }$ on $[ a , b ]$. For every parameter $\varepsilon > 0$, we set
$$J _ { \varepsilon } = \int _ { a } ^ { b } \psi \left( \frac { x } { \varepsilon } \right) f ( x ) d x$$
First case. In this question, we further assume that $\psi$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R }$ and that $f$ has compact support in $] a , b [$.\\
(a) Show that for all $\varepsilon > 0$,
$$\left| J _ { \varepsilon } - c _ { 0 } ( \psi ) \left( \int _ { a } ^ { b } f ( x ) d x \right) \right| \leq \varepsilon ( b - a ) \left\| f ^ { \prime } \right\| _ { \infty } \sum _ { n \in \mathbb { Z } ^ { * } } \frac { \left| c _ { n } ( \psi ) \right| } { | n | }$$
Hint. One can reduce to the case where $\int _ { 0 } ^ { 2 \pi } \psi ( y ) d y = 0$.\\
(b) Deduce the limit of $J _ { \varepsilon }$ as $\varepsilon \rightarrow 0$.