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In discussions of Sobolev spaces one often sees the Japanese bracket, $$\langle x \rangle = (1+|x|^2)^{1/2},$$ as useful shorthand.

I was not easily able to find information about this term.

(1) What was the first systematic usage of something like this shorthand?

(2) What is the origin of the specific phrase "Japanese bracket"?

(3) Why is it called the "Japanese bracket"? The answer to this may be obvious from the answers to either (1) or (2).

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I can provide a partial response to this question. As noted in my earlier comment, the term "Japanese bracket" appears in the 1999 paper "Global existence of small solutions to the generalized derivative nonlinear Schrödinger equation”'' (p. 135, journal pagination).

I contacted two of the authors of this paper, and received one response regarding the notation. The terminology predates the paper. The author who responded commented that he had heard the term used at several conferences prior to the drafting of the 1999 paper, and that the notation was widely used by Japanese analysts at that time. The inference is that the answer to your third question is "due to widespread use by the Japanese mathematical community". His response also provides a temporal bound for your first question: no later than the late 1990s.

If you're interested in more specifics, I'd recommend pulling on this thread a bit more by directly contacting early users of the notation / terminology.

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    $\begingroup$ The authors were Nakao Hayashia, Changxing Miaob and Pavel I. Naumkin. Who did you contact, and who responded? $\endgroup$
    – J.G.
    Commented Aug 12, 2021 at 6:28
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    $\begingroup$ I contacted Hayashi and Naumkin, and received a response from Naumkin. $\endgroup$
    – Mark Yasuda
    Commented Aug 13, 2021 at 0:42
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    $\begingroup$ This notation itself is certainly older, e.g. in the book "Pseudo-Differential Operators" by Kumano-go this notation is introduced on page xv. (The translation I know is from '82 but the original japanese version should be from around 1974.) $\endgroup$
    – Errol
    Commented Apr 20, 2022 at 12:38

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