l._ Introduction
The edge-of-the-wedge theorem deals with a question about analytic continuation
of holomorphic functions of several complex variables. The problem arose in physics,
in connection with quantum field theory and dispers ion relations, and the theorem was
first proved by Bogolyubov in this connection (see [2, p. 672]; also [3], [4], [5] , [8] ,
[ É 5 à. The present paper gives complete and self-contained proofs of several versions
of this theorem, and also contains some applications to function theory in a polydisc.
My knowledge of physics is much too small to permit me to say anything intelligent
either about the problems that led to the original discovery of the theorem, or about
the physical Information that the theorem contains.
Throughout this paper, ç will be a positive integer, Rn will denote n-dimensional
Euclidean space, and Cn will be the space of ç complex variables an n-dimension-
al vector space over the complex field C with the usual product topology. Points
of Cn will be written
ae = (ae , · · . , ae ) (z.EC)
é ç é
ae = ÷ + iy, with ÷ Rn, y Rn,
where ae . = ÷. + iy., ÷ = ^ ·»· , ÷ )9 y = (y , · · · , y ). In this way Cn is regarded
as the cartesian product of two copies of Rn. (When ç = 1, these are the real and
imaginary axes in the complex plane.) When Á CRn and  C Rn, Á + iB is the set
of all ae = ÷ + iy with x A, y B. In particular Cn = Rn + *Än.
The unit polydisc in Cn is the set Un which consists of all ae = (ae., · · · , ae )
with I ae , É 1 for 1 7 ç.
If Ù is an open set in C n , a complex-valued function / in 0 is said to be holo-
morphic in Ù if (a) / is continuous in Ù and (b) / is holomorphic in each variable
separately. Very elementary arguments (see, for instance, [9l o r [ l l ] ) show that every
holomorphic function / in Ù is locally the sum of a multiple power series. More pre-
cisely, if ae £ Ù, there exists r 0 such that ae + rUn C Ù; for every w rUn, we
r( \
/U + w)=2w—1— w
á á *
Previous Page Next Page