In this part, $\varphi : [ a , b ] \rightarrow \mathbb { R }$ and $f : [ a , b ] \rightarrow \mathbb { R }$ are two functions of class $\mathcal { C } ^ { \infty }$. We are interested in integrals of the form $$I ( \lambda ) = \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } f ( x ) d x$$ where $\lambda$ is a strictly positive real parameter. Case where the phase can be stationary. Throughout this question, we assume that $\left| \varphi ^ { \prime \prime } ( x ) \right| \geq 1$ for all $x \in [ a , b ]$. (a) Show that $\varphi ^ { \prime }$ is strictly monotone on $[ a , b ]$ and that there exists a unique point $c \in [ a , b ]$ such that $\left| \varphi ^ { \prime } ( c ) \right| = \inf _ { x \in [ a , b ] } \left| \varphi ^ { \prime } ( x ) \right|$. (b) If $x \in [ a , b ]$, show that $\left| \varphi ^ { \prime } ( x ) \right| \geq | x - c |$. (c) Show that for all $\delta > 0$, $$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x \right| \leq 2 c _ { 1 } ( \lambda \delta ) ^ { - 1 } + 2 \delta$$ (d) Deduce that there exists a constant $c _ { 2 }$, independent of $\lambda , \varphi , a$ and $b$ such that $$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x \right| \leq c _ { 2 } \lambda ^ { - 1 / 2 }$$ (e) Show that $$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } f ( x ) d x \right| \leq c _ { 2 } \lambda ^ { - 1 / 2 } \left( | f ( b ) | + \int _ { a } ^ { b } \left| f ^ { \prime } ( x ) \right| d x \right)$$
In this part, $\varphi : [ a , b ] \rightarrow \mathbb { R }$ and $f : [ a , b ] \rightarrow \mathbb { R }$ are two functions of class $\mathcal { C } ^ { \infty }$. We are interested in integrals of the form
$$I ( \lambda ) = \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } f ( x ) d x$$
where $\lambda$ is a strictly positive real parameter.
Case where the phase can be stationary. Throughout this question, we assume that $\left| \varphi ^ { \prime \prime } ( x ) \right| \geq 1$ for all $x \in [ a , b ]$.\\
(a) Show that $\varphi ^ { \prime }$ is strictly monotone on $[ a , b ]$ and that there exists a unique point $c \in [ a , b ]$ such that $\left| \varphi ^ { \prime } ( c ) \right| = \inf _ { x \in [ a , b ] } \left| \varphi ^ { \prime } ( x ) \right|$.\\
(b) If $x \in [ a , b ]$, show that $\left| \varphi ^ { \prime } ( x ) \right| \geq | x - c |$.\\
(c) Show that for all $\delta > 0$,
$$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x \right| \leq 2 c _ { 1 } ( \lambda \delta ) ^ { - 1 } + 2 \delta$$
(d) Deduce that there exists a constant $c _ { 2 }$, independent of $\lambda , \varphi , a$ and $b$ such that
$$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x \right| \leq c _ { 2 } \lambda ^ { - 1 / 2 }$$
(e) Show that
$$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } f ( x ) d x \right| \leq c _ { 2 } \lambda ^ { - 1 / 2 } \left( | f ( b ) | + \int _ { a } ^ { b } \left| f ^ { \prime } ( x ) \right| d x \right)$$