6
WALTER RUDIN
not seem to exist in any of the Standard texts. In fact, the passage from Á to  is not
really any easier when ç = 1 than when ç 1.
3. Tubes with a common edge
Any set of the form Rn + iV, where V is an open subset of Rn, is called an
open tube in Cn. For instance, when ç = I the region bounded by any pair of lines
parallel to the real axis is a tube.
The convex hüll of a set Ì in Rn is, by definition, the smallest eonvex subset
of Rn that contains M. If Ì is open, so is its convex hüll, as can be seen very
easily.
We now present an analogue of Theorem Á which is more special in one sense
but is more general in another. The special feature is that the edge Å is now Rn; also
(but this is les s important) a growth condition is imposed on /. The added generality
is that the sets F and V that go into the hypothesis of Theorem Á are replaced by
almost arbitrary open sets Vl and V2- In particular, they have nothing to do with
cones.
Theorem. Suppose V. and F . are bounded connected open sets in Rn9 both of
which have the origin as á boundary point* Put V = V. U V'. Assume
(i) / is holomorphic in Rn + iV,
(ii) / is continuous on Rn + iV,
(iii) there is á constant Á oo such that
| / U + iy)\ expM(%2
+
. . .
+
*2)j
in Rn + iV.
Then f extends to á holomorphic function in Rn + iW, where*W is the convex
hüll of F. This extension is continuous on Rn + iW.
The given / is holomorphic in each of the tubes Rn + iVt and Rn + iV2 Note
that Rn + iW is the convex hüll of the union of these tubes.
The proof uses certain convexity arguments. It seems best to State these sep-
arately as a lemma. We shall use the scalar product notation
÷.*=£vf
z'ß=Óí/
for t Rn, ÷ Rn, ae Å Cn. Also, for y Rn, e will be the function defined on Rn
by e (t ) = e"y't. The Symbols L , L2 denote the usual Spaces of integrable functions,
relative to Lebesgue measure on Rn.
Lemma. Suppose u L , and —» ue is á continuous map of some compact set
KCRn into L1. Then (a) and (b) are true:
(a) y —* ue is á continuous map of Ê into L , where Ê is the convex hüll of K.
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