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We continue with the notation of the previous section.
Using the order structure on
we define
some classes of analytic functions on
:
and
(16) |
 |
We clearly have
We can now give the definition of analytic measures in
.
Definition 3.1
Let
be a sup path attaining representation
of
by isomorphisms of
.
A measure
is
called weakly analytic if the mapping
is in
for every
.
That the two definitions of analyticity are
equivalent will be shown later in this section.
Since
is translation-invariant,
it follows readily that
for all
,
and hence
(18) |
 |
We now recall several
basic results from
spectral theory of bounded functions that will be needed in the sequel. Our reference is [21, Section 40].
If
is in
, write
for the smallest weak-* closed translation-invariant subspace of
containing
, and let
denote the closed translation-invariant ideal in
:
It is clear that
.
The spectrum of
, denoted by
,
is the set of all continuous characters of
that belong to
.
This closed subset of
is also given by
(19) |
![$\displaystyle \sigma \left[\phi\right]=Z({\cal I}(\phi)).$](img168.png) |
(See [21, Theorem (40.5)].)
Recall that a closed subset
of
is
a set of spectral synthesis for
, or an
-set, if and only if
is the only ideal in
whose zero set
is
.
There are various equivalent definitions of
-sets.
Here is one that we will use at several occasions.
A set
is an
-set if and only if
every essentially bounded function
in
with
is the weak-* limit
of linear combinations of characters from
.
(See [21, (40.23) (a)].) This has
the following immediate consequence.
Proof. Part (i) is a simple consequence of
[21, Theorems (40.8) and (40.10)].
We give a proof for the sake of completeness.
Write
as the weak-* limit of
trigonometric polynomials,
,
with characters in
. Then
since
vanishes on
.
To prove (ii), assume that
is not 0 a.e..
Then, there is
in
such that
is not 0 a.e.. But this contradicts (i),
since
,
is in
,
and
on
.
The following is a converse of sorts of Proposition
3.3 and follows easily from definitions.
Proof. Let
be any
element in
. We will show that
is not in the spectrum of
by constructing a
function
in
with
and
. Let
be an open neighborhood of
not intersecting
, and let
be in
such that
is equal to 1 at
and to 0 outside
.
Direct computations show that the Fourier transform
of the function
, when evaluated at
, gives
,
and hence it vanishes on
.
It follows from (20) that
,
which completes the proof.
A certain class of
-sets, known as the
Calderón sets, or
-sets,
is particularly useful to us.
These are defined as follows.
A subset
of
is
called a
-set if every
in
with Fourier transform
vanishing on
can be approximated
in the
-norm by functions of the
form
where
and
vanishes on
an open set containing
.
-sets enjoy the following properties
(see [21, (39.39)] or [22, Section 7.5]).
- Every
-set is an
-set.
- Every closed subgroup of
is a
-set.
- The empty set is a
-set.
- If the boundary of a set
is a
-set,
then
is a
-set.
- Finite unions of
-sets are
-sets.
Since closed subgroups are
-sets, we conclude that
,
and
, for all
,
are
-sets.
>From the definition of
, (13),
and the fact that
is open and closed, it
follows that the boundary of
is the
closed subgroup
.
Hence
is a
-set.
For
, the set
is open and closed,
and so it has
empty boundary, and thus it is a
-set. Likewise
is a
-set for all
.
>From this we conclude that
arbitrary unions of
and
are
-sets, because an arbitrary union of
such sets, not including the index
, is open and closed,
and so it is a
-set.
We summarize our findings as follows.
Proposition 3.5
Suppose that
is a measurable order on
.
We have:
(i)
and
are
-sets;
(ii)
is a
-set for all
;
(iii) arbitrary unions of
and
are
-sets.
As an immediate application, we
have the following characterizations.
Corollary 3.6
Suppose that
is in
,
then
(i)
if and only if
for all
such that
on
;
(ii)
if and only if
;
(iii)
if and only if
.
Proof. Assertions (i) and (iii) are clear from
Propositions 3.5 and 3.4.
To prove
(ii), use Fubini's Theorem to first establish
that for any
, and any
, we have
Now suppose that
, and let
be any function in
. From Propositions 3.5 and
3.4, we have that
for all
with Fourier transform vanishing on
, equivalently, for all
.
Hence,
for all
in
, from which it follows that
.
The converse is proved similarly, and we omit the details.
Aiming for a characterization of weakly analytic measures in terms of their spectra, we present one more result.
Proposition 3.7
Let
be weakly measurable in
.
(i) Suppose that
is a nonvoid closed
subset of
and
.
Then
for all
.
(ii) Conversely, suppose that
is an
-set in
and that
for all
, then
.
Proof. We clearly have
.
Hence,
and (i) follows.
Now suppose that
is an
-set and let
be such that
on
. Then, for all
,
we have from Proposition 3.4:
Equivalently, we
have that
Since the Fourier transform of the function
vanishes on
,
we see that
.
Thus
,
which completes the proof.
Straightforward applications of Propositions
3.5
and 3.7 yield the
desired characterization of weakly analytic measures.
Corollary 3.8
Suppose that
. Then,
(i)
is weakly
analytic if and only if
if and only if
, for every
;
(ii)
if and only if
for every
.
(iii)
if and only if
for every
.
(iv)
if and only if
for every
.
The remaining results of this section are
simple properties of measures in
that will be needed later.
Although the statements are
direct analogues of classical facts about measures on groups, these
generalization require in some places the sup path
attaining property of
.
Proposition 3.9
Suppose that
and
.
Then
is contained in the support
of
, and
.
Proof.
Given
not in the support of
, to
conclude that it is also not in the spectrum of
it is enough to find a function
in
with
and
. Simply choose
with Fourier transform
vanishing on the support of
and taking value 1
at
. By Fourier inversion, we have
, and since
, the
first part of the proposition follows.
For the second part,
we have
,
which implies the desired inclusion.
We next prove a property of
functions
similar to the characterization of
functions which are
constant on cosets of a subgroup [21, Theorem (28.55)].
Proposition 3.10
Suppose that
is in
and that
is
an open subgroup of
. Let
denote
the normalized Haar measure on the compact group
, the annihilator in
of
(see [20, (23.23)]. Then,
if and only if
a. e. This is also
the case if and only if
is constant on
cosets of
.
Proof. Suppose that the spectrum of
is contained
in
. Since
is an
-set, it follows that
is the weak-* limit of trigonometric polynomials with
spectra contained in
. Let
be a net
of such trigonometric polynomials converging
to
weak-*. Note that, for any
,
we have
. For
in
, we have
In particular, we have
and so
Since this holds for any
in
, we conclude that
converges weak-* to
.
But
, and
converges weak-* to
, hence
. The remaining assertions of the
lemma are easy to prove. We omit the details.
In what follows, we use the symbol
to denote the
normalized Haar measure on the compact subgroup
, the annihilator in
of
.
This measure is also characterized by its Fourier
transform:
(see [20, (23.19)]).
Corollary 3.11
Suppose that
. Then,
(i)
if and only if
;
(ii)
if and only if
.
Proof. (i) If
,
then, by Proposition 3.10,
.
Hence by Corollary 3.8,
.
For the other direction, suppose that
. Then by Corollary 3.8
we have that the spectrum of the function
is contained in
for every
. By Proposition 3.10,
we have that
for almost all
. Since this holds for all
,
the desired conclusion
follows from Proposition 1.4.
Part (ii) follows from Corollary 3.6 (ii), Proposition
3.7(ii), and the fact that
is an
-set.
Corollary 3.12
Suppose that
and
,
and let
. Then
.
Proof. For any
, we have from Corollary
3.11
Next: Homomorphism theorems
Up: Decomposition of analytic measures
Previous: Orders on locally compact
Stephen Montgomery-Smith
2002-10-30