What does the notation:
$$O_{\mathcal{S}'\rightarrow \mathcal{S}}(h^\infty)$$
mean in Zworski's semiclassical analysis?
(To be clear, I know what $O(h^\infty)$ means in the context of normed and semi-normed spaces.)
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That notation would arise when you're looking at the asymptotics of a linear operator parametrized in $h$. In particular, for your case this means that the asymptotic formula has an error operator $E_h$, such that, for all $u \in \mathscr{S}'$, and for multi-indices $\alpha, \beta$, and integer $N > 0$, there exists $C_{\alpha, \beta, N,u} > 0$ such that $\| E_h u\|_{\alpha, \beta} \le C_{\alpha, \beta, N,u} h^N$ for sufficiently small $h$, where $\| \cdot \|_{\alpha, \beta}$ is the usual semi-norm for the Schwartz space.
Christopher A. Wong
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And the $O(h^{\infty})$? I had never seen that one and don't know how to interpret it. I mean, the $h^{\infty}$ part. – OR. Aug 02 '13 at 01:10
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Yes, this is how I interpret such notation when the domain and codomain are normed spaces, but this is not the case here...how can we assert the existence of an operator norm? @RGB $O(h^\infty)$ means $O(h^n)$ for all positive integers $n$ as $h$ gets small. – goonfiend Aug 02 '13 at 02:12
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@SeanGomes, I'm sorry, I see that I replied in haste. I'll modify my answer to include an extension of this notation to seminormed spaces. – Christopher A. Wong Aug 02 '13 at 04:05
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Okay thankyou, I wasn't sure if it was a pointwise bound or something more uniform. Perhaps a generalisation of the UBP to locally convex vector spaces makes this distinction moot anyway. – goonfiend Aug 02 '13 at 04:28