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 $$J _ { \varepsilon } = \int _ { a } ^ { b } \psi \left( \frac { x } { \varepsilon } \right) f ( x ) d x$$ Second case. We now assume only that $\psi \in \mathcal { C } ^ { 0 } ( \mathbb { R } )$ is periodic with period $2 \pi$, and $f \in \mathcal { C } ^ { 1 } ( [ a , b ] )$. Let $\varepsilon > 0$. We define a subdivision of the interval $[ a , b ]$ as follows. We denote $N _ { \varepsilon }$ the integer part of $\frac { b - a } { 2 \pi \varepsilon }$. We then define $$x _ { k } ^ { \varepsilon } = a + 2 k \pi \varepsilon , \text { for every integer } k \text { such that } 0 \leq k \leq N _ { \varepsilon } .$$ (a) Show that $\lim _ { \varepsilon \rightarrow 0 } x _ { N _ { \varepsilon } } ^ { \varepsilon } = b$. (b) Deduce that $$\lim _ { \varepsilon \rightarrow 0 } \int _ { x _ { N _ { \varepsilon } } ^ { \varepsilon } } ^ { b } \psi \left( \frac { x } { \varepsilon } \right) f ( x ) d x = 0$$ (c) Show that for every integer $k$ such that $0 \leq k \leq N _ { \varepsilon } - 1$, for all $x \in \left[ x _ { k } ^ { \varepsilon } , x _ { k + 1 } ^ { \varepsilon } \right]$, $$\left| f ( x ) - f \left( x _ { k } ^ { \varepsilon } \right) \right| \leq 2 \pi \varepsilon \left\| f ^ { \prime } \right\| _ { \infty }$$ (d) Show that $$\sum _ { k = 0 } ^ { N _ { \varepsilon } - 1 } \int _ { x _ { k } ^ { \varepsilon } } ^ { x _ { k + 1 } ^ { \varepsilon } } \psi \left( \frac { x } { \varepsilon } \right) f \left( x _ { k } ^ { \varepsilon } \right) d x = \left( \int _ { 0 } ^ { 2 \pi } \psi ( y ) d y \right) \left( \varepsilon \sum _ { k = 0 } ^ { N _ { \varepsilon } - 1 } f \left( x _ { k } ^ { \varepsilon } \right) \right)$$ (e) Show that $$\left| \sum _ { k = 0 } ^ { N _ { \varepsilon } - 1 } \int _ { x _ { k } ^ { \varepsilon } } ^ { x _ { k + 1 } ^ { \varepsilon } } \psi \left( \frac { x } { \varepsilon } \right) \left( f ( x ) - f \left( x _ { k } ^ { \varepsilon } \right) \right) d x \right| \leq \varepsilon ( b - a ) \left\| f ^ { \prime } \right\| _ { \infty } \left( \int _ { 0 } ^ { 2 \pi } | \psi ( y ) | d y \right)$$ (f) Deduce that $\lim _ { \varepsilon \rightarrow 0 } J _ { \varepsilon } = \left( \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \psi ( y ) d y \right) \left( \int _ { a } ^ { b } f ( x ) d x \right)$.
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
$$J _ { \varepsilon } = \int _ { a } ^ { b } \psi \left( \frac { x } { \varepsilon } \right) f ( x ) d x$$
Second case. We now assume only that $\psi \in \mathcal { C } ^ { 0 } ( \mathbb { R } )$ is periodic with period $2 \pi$, and $f \in \mathcal { C } ^ { 1 } ( [ a , b ] )$. Let $\varepsilon > 0$. We define a subdivision of the interval $[ a , b ]$ as follows. We denote $N _ { \varepsilon }$ the integer part of $\frac { b - a } { 2 \pi \varepsilon }$. We then define
$$x _ { k } ^ { \varepsilon } = a + 2 k \pi \varepsilon , \text { for every integer } k \text { such that } 0 \leq k \leq N _ { \varepsilon } .$$
(a) Show that $\lim _ { \varepsilon \rightarrow 0 } x _ { N _ { \varepsilon } } ^ { \varepsilon } = b$.\\
(b) Deduce that
$$\lim _ { \varepsilon \rightarrow 0 } \int _ { x _ { N _ { \varepsilon } } ^ { \varepsilon } } ^ { b } \psi \left( \frac { x } { \varepsilon } \right) f ( x ) d x = 0$$
(c) Show that for every integer $k$ such that $0 \leq k \leq N _ { \varepsilon } - 1$, for all $x \in \left[ x _ { k } ^ { \varepsilon } , x _ { k + 1 } ^ { \varepsilon } \right]$,
$$\left| f ( x ) - f \left( x _ { k } ^ { \varepsilon } \right) \right| \leq 2 \pi \varepsilon \left\| f ^ { \prime } \right\| _ { \infty }$$
(d) Show that
$$\sum _ { k = 0 } ^ { N _ { \varepsilon } - 1 } \int _ { x _ { k } ^ { \varepsilon } } ^ { x _ { k + 1 } ^ { \varepsilon } } \psi \left( \frac { x } { \varepsilon } \right) f \left( x _ { k } ^ { \varepsilon } \right) d x = \left( \int _ { 0 } ^ { 2 \pi } \psi ( y ) d y \right) \left( \varepsilon \sum _ { k = 0 } ^ { N _ { \varepsilon } - 1 } f \left( x _ { k } ^ { \varepsilon } \right) \right)$$
(e) Show that
$$\left| \sum _ { k = 0 } ^ { N _ { \varepsilon } - 1 } \int _ { x _ { k } ^ { \varepsilon } } ^ { x _ { k + 1 } ^ { \varepsilon } } \psi \left( \frac { x } { \varepsilon } \right) \left( f ( x ) - f \left( x _ { k } ^ { \varepsilon } \right) \right) d x \right| \leq \varepsilon ( b - a ) \left\| f ^ { \prime } \right\| _ { \infty } \left( \int _ { 0 } ^ { 2 \pi } | \psi ( y ) | d y \right)$$
(f) Deduce that $\lim _ { \varepsilon \rightarrow 0 } J _ { \varepsilon } = \left( \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \psi ( y ) d y \right) \left( \int _ { a } ^ { b } f ( x ) d x \right)$.