 
Summary: 1
MULTIPLIERS AND A PALEYWIENER THEOREM
FOR REAL REDUCTIVE GROUPS
James Arthur
The classical PaleyWiener theorem is a description of the image
of Cc(OR) under Fourier transform. The Fourier transform
f(A) = f(x)eAX dx
00
is defined a priori for purely imaginary numbers A, but if f has
compact support f will extend to an entire function on the complex
plane. The image of C((R) under this map is the space of entirec
functions F with the following property  there exists a constant
N such that
supflF(A) eNIRe Al(1 + lIm A)n] <
AeC
for every integer n. (There is a similar theorem which characterizes
the image of the space of compactly supported distributions.)
Our purpose is to describe an analogous result for a reductive
Lie group. We shall also discuss a closely related theorem on
multipliers, a result whose statement is especially simple. Both
