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 of a non-stationary phase. We assume in this question that $\varphi ^ { \prime } ( x ) \neq 0$ for all $x \in [ a , b ]$.
(a) We define $L : \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } ) \rightarrow \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$ and $M : \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } ) \rightarrow \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$ by: for all $g \in \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$, all $x \in [ a , b ]$, $$L g ( x ) = \frac { 1 } { i \lambda \varphi ^ { \prime } ( x ) } g ^ { \prime } ( x ) , \quad M g ( x ) = - \left( \frac { g } { i \varphi ^ { \prime } } \right) ^ { \prime } ( x )$$ i. Determine the functions $g \in \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$ such that $L g = g$. ii. Let $g , h \in \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$. We assume that $h$ has compact support in $] a , b [$. Show that $$\int _ { a } ^ { b } h ( x ) L g ( x ) d x = \frac { 1 } { \lambda } \int _ { a } ^ { b } g ( x ) M h ( x ) d x$$ (b) Show that if $f$ has compact support in $] a , b [$, then for all $N \in \mathbb { N } ^ { * }$, there exists a constant $\gamma _ { N }$ independent of $\lambda$ such that $$| I ( \lambda ) | \leq \gamma _ { N } \lambda ^ { - N }$$