1. ABSTRAC T HARMONIC ANALYSIS: A SURVEY 13

Setting g = 0 we obtain the Poisson summation formula:

(PSF) $o = £*(')•

This formula will be valid for $ of Poisson type with respect to the pair (G, L) and

with a Haar measure d\i such that vol(G/L) = 1.

We now make a remark that will play an important role in the number theoretic

situation. Suppose $ is a function of Poisson type with respect to the pair (G, L)

for which at least one side of (PSF) is not 0. Call dp the dual measure to d/i; put

JQ/L±

dp = c, and perform the calculation leading to (PSF) with the roles of G

and G interchanged, starting with the Fourier transform of $ with respect to the

Haar measure

c~ldp

on G; by the inversion formula this is c

- 1

$ , and we get the

same Poisson formula with $ replaced by c_1I. By comparing the two formulas we

get c = 1. It follows that the measures d\i and dp on G and G, normalized so that

vol^(G/L) — voljx{G/L^) = 1 are dual to each other. In the number theoretic case

of interest, where G is identified with G by means of an isomorphism which sends

L onto

L-1,

the self-dual measure d\i on G will be that for which voll_i(G/L) — 1.

Our basic examples of functions of Poisson type will come from spaces of

Schwartz-Bruhat functions.

Measures on Restricted Direct Products

Let G be the restricted direct product of the locally compact groups {Gv}ves

with respect to the subgroups {Hv}vesi where the Hv are compact for almost all

v. For each v let dfiv be a measure on Gv such that JH d\xv — 1 for almost all v.

We now show how to associate with this data a unique measure on G. Since by

definition G is the inductive limit of open subgroups of the form

Gs=l[Gvx Y[HV,

vES v£S

where 5 is a finite subset of S, it suffices to construct a family of coherent measures

djis on these subgroups. The choice of measure on Gs is

ves

where

d/js

is that measure on the factor

Gs

= Ylvds Hv for which

/ dfis = TT / d/iv (a finite product!).

To verify that the dfis are coherent we must show that for any two finite sets T and

S with T D S, in which case GT 3 Gs, the restriction of dfir to Gs coincides with

df^s- It is clearly sufficient to verify this for two subsets T and S with T — SU{w}.

Now, since

Gr = ±lw X Gr ,

we have that

dfis

— d\i x

d/iT

is the required measure on

Gs.

We have therefore

dfis = TT dfiv x dfiS