TITLE: Pseudo-differential operators
QUESTION [5 upvotes]: What is the meaning of the formula $\sigma (PQ)=\sum \frac{1}{\alpha!}\partial _{\xi }^{\alpha}pD_{x}^{\alpha}q\; ;\;\;  \sigma (Q)=q,\;\;\; \sigma (P)=p$ 
if the series on right side is infinite?
Thanks a lot!

REPLY [2 votes]: The equality sign is irrelevant. It's actually the asymptotic relation
$$
\sigma_{AB}(x,\xi)\sim\sum_\alpha\frac{1}{\alpha!}\partial^\alpha_{\xi}\sigma_A(x,\xi)\cdot \partial^\alpha_x\sigma_B(x,\xi)
$$
where the sum is taken over all possible multi-indices $\alpha\in\mathbb{Z}_+^n$ if $(x,\xi)\in\mathbb{R}^n\times\mathbb{R}^n$.
By definition, a symbol $\sigma$ is asymptotic to a series $\sum_{j=1}^\infty{\sigma_j}$ where $\sigma_j\in S^{m_j}(\Omega),\;\Omega\subset\mathbb{R}^n$ and $m_j\rightarrow -\infty$ iff for each N the following relation is satisfied:
$$
(\sigma-\sum_{j=1}^N\sigma_j )\in S^{\overline{m_{N+1}}}(\Omega),\;\overline{m_k}=\max[m_j\mid j\geq k].
$$
It follows easily that two symbols asymptotic to the same series may differ by not more than a symbol of the class $S^{-\infty}(\Omega)$.