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Suppose that some smooth function $f \in C^\infty\bigl(\mathbb R^n \times (0,+\infty)\bigr)$ possesses an asymptotic development $$ f(x,t) \sim t^{-\alpha} e^{ith(x)} \sum\limits_{k=0}^{+\infty} a_k(x)t^{-k}, \quad t \to + \infty, \;\text{uniformly w.r.t. $x$} $$ Here $h$ is some real-valued smooth function and $\alpha>0$. Suppose further that functions $$ F_k(t) = \int\limits_{\Omega} e^{it h(x)}a_k(x) \, dx, \quad \Omega \Subset \mathbb R^n, $$ possess their own asymptotic developments $$ F_k(t) \sim t^{-\alpha} e^{i t \lambda } \sum\limits_{l=0}^\infty b_{k,l} t^{-l}, \quad t \to + \infty, \;\; \lambda \in \mathbb R, \;\; k \in \mathbb N \cup \{0\}. $$ Is it possible to say then that function $F(t) = \int_\Omega f(x,t) \, dx$ has some natural asymptotic development? In the case $\alpha = 0$ it is true but in the case $\alpha > 0$ I don't see which asymptotic sequence I should take.

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You have that $$ \left| {F\left( t \right) - \sum\limits_{k = 0}^{N - 1} {F_k \left( t \right)t^{ - k - \alpha } } } \right| = \left| {\int_\Omega {\left( {f\left( {x,t} \right) - \sum\limits_{k = 0}^{N - 1} {a_k \left( x \right)e^{ith\left( x \right)} t^{ - k - \alpha } } } \right)dx} } \right| $$ $$ \le \int_\Omega {\left| {f\left( {x,t} \right) - \sum\limits_{k = 0}^{N - 1} {a_k \left( x \right)e^{ith\left( x \right)} t^{ - k - \alpha } } } \right|dx} \le \int_\Omega {C_N \left| {e^{ith\left( x \right)} } \right|t^{ - N - \alpha } dx} = {\mathop{\rm Vol}\nolimits} \left( \Omega \right)C_N t^{ - N - \alpha } , $$ for large enough $t$ and bounded $\Omega$. So you have $$ F\left( t \right) = \sum\limits_{k = 0}^{N - 1} {F_k \left( t \right)t^{ - k - \alpha } } + \mathcal{O}\left( {t^{ - N - \alpha } } \right), $$ for large $t$ and any fixed $N$. Substitute truncated asymptotic series for the $F_k(t)$'s and collect the error term to show that $F(t)$ has an asymptotic expansion.

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