Malgrange–Zerner theorem

In mathematics, Malgrange–Zerner theorem (named for Bernard Malgrange and Martin Zerner) shows that a function on allowing holomorphic extension in each variable separately can be extended, under certain conditions, to a function holomorphic in all variables jointly. This theorem can be seen as a generalization of Bochner's tube theorem to functions defined on tube-like domains whose base is not an open set.

Theorem[1][2] Let

and let convex hull of . Let be a locally bounded function such that and that for any fixed point the function is holomorphic in in the interior of for each . Then the function can be uniquely extended to a function holomorphic in the interior of .

History

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According to Henry Epstein,[1][3] this theorem was proved first by Malgrange in 1961 (unpublished), then by Zerner [4] (as cited in [1]), and communicated to him privately. Epstein's lectures [1] contain the first published proof (attributed there to Broz, Epstein and Glaser). The assumption   was later relaxed to   (see Ref.[1] in [2]) and finally to  .[2]

References

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  1. ^ a b c d Epstein, Henry (1966). Some analytic properties of scattering amplitudes in quantum field theory (8th Brandeis University Summer Institute in Theoretical Physics: Particle symmetries and axiomatic field theory). pp. 1–128.
  2. ^ a b c Drużkowski, Ludwik M. (1999-02-22). "A generalization of the Malgrange–Zerner theorem". Annales Polonici Mathematici. 38 (2): 181–186. doi:10.4064/ap-38-2-181-186. Retrieved 2021-07-01.
  3. ^ Epstein, H. (1963). "On the Borchers class of a free field" (PDF). Il Nuovo Cimento. 27 (4): 886–893. doi:10.1007/bf02783277. S2CID 120708058.
  4. ^ Zerner M. (1961), mimeographed notes of a seminar given in Marseilles