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. (a) We assume that $\left| \varphi ^ { \prime } ( x ) \right| \geq 1$ for all $x \in [ a , b ]$ and that $\varphi ^ { \prime }$ is monotone on $[ a , b ]$. Show that there exists a constant $c _ { 1 } > 0$, independent of $\lambda , \varphi$ and of $a , b$, such that $$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x \right| \leq c _ { 1 } \lambda ^ { - 1 }$$ Hint. One can write $$\int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x = \int _ { a } ^ { b } i \lambda \varphi ^ { \prime } ( x ) e ^ { i \lambda \varphi ( x ) } \frac { 1 } { i \lambda \varphi ^ { \prime } ( x ) } d x$$ and integrate by parts. (b) Let $\delta > 0$. We assume that $\left| \varphi ^ { \prime } ( x ) \right| \geq \delta$ for all $x \in [ a , b ]$ and that $\varphi ^ { \prime }$ is monotone on $[ a , b ]$. Show that $$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x \right| \leq c _ { 1 } ( \lambda \delta ) ^ { - 1 }$$
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.
(a) We assume that $\left| \varphi ^ { \prime } ( x ) \right| \geq 1$ for all $x \in [ a , b ]$ and that $\varphi ^ { \prime }$ is monotone on $[ a , b ]$. Show that there exists a constant $c _ { 1 } > 0$, independent of $\lambda , \varphi$ and of $a , b$, such that
$$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x \right| \leq c _ { 1 } \lambda ^ { - 1 }$$
Hint. One can write
$$\int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x = \int _ { a } ^ { b } i \lambda \varphi ^ { \prime } ( x ) e ^ { i \lambda \varphi ( x ) } \frac { 1 } { i \lambda \varphi ^ { \prime } ( x ) } d x$$
and integrate by parts.\\
(b) Let $\delta > 0$. We assume that $\left| \varphi ^ { \prime } ( x ) \right| \geq \delta$ for all $x \in [ a , b ]$ and that $\varphi ^ { \prime }$ is monotone on $[ a , b ]$. Show that
$$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x \right| \leq c _ { 1 } ( \lambda \delta ) ^ { - 1 }$$