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 Â 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 Ã 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.)