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I have trouble making sense of the following remark that is given in a set of lecture notes I am currently going through on my own:

" Oscillatory integrals are used for the study of singularities of functions and distributions, and therefore all calculations are carried out modulo $S^{\, -\infty}$."

The oscillatory integral is given in my notes by

\begin{equation} \mathcal{I}_a(x,y) = (2\pi)^{-n} \int e^{i(x-y) \cdot \xi} \ a(x,y,\xi) \ d\xi \end{equation}

I understand that we write $a(x,y,\xi) \in S^{\ m}$ for $m \in \mathbb{R}$ if

$a(x,y,\xi)$ is $C^\infty$ on $\mathbb{R}^n_x \times \mathbb{R}^n_y \times \mathbb{R}^n_\xi$

and if $a$ satsfies \begin{equation} |\partial^\alpha_\xi \partial^\beta_x \partial^\gamma_y a(x,y,\xi)| \leq \text{Const}_{\alpha, \beta, \gamma}(1 + |\xi|)^{m - |\alpha|} \end{equation} for all multi - indices $\alpha, \beta, \gamma$

Then we define $S^{-\infty} = \cap_m S^{-m}$ where $m \in \mathbb{R}$.

But how can I formally write by what is meant by carrying out calculations modulo $S^{- \infty}$ ?

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