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

or

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 Â 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

have

r( \

TrWaf)iz)

a

/U + w)=2w—1— w

á á *

1