4
WALTER RUDIN
paper does not depend on § 3 . One reason for including it is that it gives a goodheuris-
tic basis for the other versions of the edge-of-the-wedge theorem. Another is that pre-
cise Information about the maximal Ù i s obtained in this case. And a third one is
simply that it is always good to have a variety of techniques that can be brought to
bear on a problem.
§ 4 contains a very neat and ingenious proof of Theorem Á that I found in [14].
In all previous papers on this subject, the passage from the continuous case to
the distribution case (i.e., from Theorem Á to Theorem B) involves a stiff dose of
rather sophisticated functional analysis. This is cut down to practically nothing in
the present paper. All that is needed i s a classical theorem of Banach and Steinhaus
whose proof— in the appropriate setting —is given in § 5 . § 6 contains some background
about the test functions ö that occur. §7 contains a lemma about the radius of con-
vergence of multiple power series which may be of some independent interest. It leads
to an easy proof of Theorem B, in §8 . The reflection theorem is proved in § 9 , and
§10 contains several applications.
Theorems Á and  deal with holomorphic functions defined in the union of two
wedges whose imaginary parts V and V are symmetrically located with respect to
the origin of Rn. It is possible to replace the cones S and S by two arhitrary open
cones in Rnf and to get a füll analogue of Theorem B. The main difference is that
the open set Ù that occurs in the conclusion need no longer contain the common
edge £ . This generalization (due to H. Epstein [â]) occupies the final §11.
2. The one-variable ease
When ç = 1, the geometric Situation described in the Introduction is particularly
simple. Å is now an open set on the real axis which, without significant loss of gen-
erality, may be taken to be a segment. In that case, W and W" are open rectangles
which have £ as a common edge, and Theorem Á says: / / / is continuous on
Å U W"~ and holomorphic in W W~ then f is actually holomorphic in Ù = W U EÜW"*
Note that this theorem does not extend the domain of /. It merely asserts that the
edge £ is a removable singularity of /. It is only when ç 1 that the remarkable
phenomenon of the enlargement of the domain occurs.
Actually, the edge £ can be replaced by any rectifiable are Ã:
Theorem. Suppose Ù is á bounded plane region whose {positively oriented)
boundary du is á rectifiable Jordan curve, Suppose Ù is divided into two regions Q
and Ù
2
by á rectifiable are à which lies in Ù except for its two end-points. Suppose
f is continuous on the closure of Ù and holomorphic in Ù- õ Ù
2
· Then f is holomor-
phic in Ù.
(For ç 1, no such generalizations of Theorem Á seem to have been proved so
far.)
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