Malgrange–Zerner theorem
In mathematics, Malgrange–Zerner theorem (named for Bernard Malgrange and Martin Zerner) shows that a function on formula_1 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 Let and let formula_3 convex hull of formula_4. Let formula_5 be a locally bounded function such that formula_6 and that for any fixed point formula_7 the function formula_8 is holomorphic  in formula_9 in the interior of formula_10 for each formula_11. Then the function formula_12 can be uniquely extended to a function holomorphic in the interior of formula_13. History. According to Henry Epstein, this theorem was proved first by Malgrange in 1961 (unpublished), then by Zerner (as cited in ), and commmunicated to him privately. Epstein's lectures contain the first published proof (attributed there to Broz, Epstein and Glaser). The assumption formula_6 was later relaxed to formula_15 (see Ref. in ) and finally to formula_16.