What is the origin of the "Japanese bracket"?

Question

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).

There is no trace of it online before 2005, when a couple of papers mention it, e.g. Stefanov-Kevrekidis, p.1843. In 2006 Tao mentioned it in Global behaviour of nonlinear dispersive and wave equations, p.29, and later promoted it on his blog, from where it likely spread. Neither bothered to comment on its origins beyond Tao saying that is "known in PDE". — Conifold, Feb 20 '21 at 05:28
The 1999 paper ``Global existence of small solutions to the generalized derivative nonlinear Schrödinger equation'' refers to the Japanese bracket on page 135. Nakao Hayashi appears to have authored some papers around this time in which this term appeared. — Mark Yasuda, Feb 20 '21 at 22:23

score 7 · Accepted Answer · answered Aug 12 '21 at 03:09

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.

The authors were Nakao Hayashia, Changxing Miaob and Pavel I. Naumkin. Who did you contact, and who responded? — J.G., Aug 12 '21 at 06:28
I contacted Hayashi and Naumkin, and received a response from Naumkin. — Mark Yasuda, Aug 13 '21 at 00:42
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.) — Errol, Apr 20 '22 at 12:38

What is the origin of the "Japanese bracket"?

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