Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $S^{- \infty}$ ?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.