Table Of ContentDiscretization of continuous frame
1 A. Fattahi, H. Javanshiri
1
0
2
n Abstract
a
J Inthispaperweconsideronthenotionofcontinuousframeofsubspaceand
3 defineanewconceptofcontinuousframe,entitledcontinuous atomic resolution
2
of identity, for arbitrary Hilbert space which has a countable reconstruction
H
] formula. Among the other result, we characterizethe relationshipbetween this
A
new concept and other known continuous frame. Finally, we state and prove
F 1
the assertions of the stability of perturbation in this concept.
.
h
t
a 1 Introduction and Preliminaries
m
[
As we know frames are more flexible tools to translate information than bases, and
1
so they are suitable replacement for bases in a Hilbert space . Finding a represen-
v
H
7 tation of f as a linear combination of the elements in frames, is the main goal
7 ∈ H
of discrete frame theory. But in continuous frame, which is a natural generalization
3
4 from discrete, itis notstraightforward. However, oneof theapplications of framesis
.
1 in wavelet theory. The practical implementation of the wavelet transform in signal
0
processing requires the selection of a discrete set of points in the transform space.
1
1 Indeed, all formulas must generally be evaluated numerically, and a computer is an
:
v intrinsically discrete object. But this operation must be performed in such a way
i
X that no information is lost. So efforts have been done to find methods to discretize
r classical continuous frames for use in applications like signal processing, numerical
a
solution of PDE, simulation, and modelling; see for example [1, 8]. In particular,
the discrete wavelet transform and Gabor frames are prominent examples and have
been proven to be a very successful tool for certain applications. Since the problem
of discretization is so important it would be nice to have a general method for this
purpose. For example, Ali, Antoine, and Gazeau in [1] asked for conditions which
ensurethatacertainsamplingofacontinuousframe ψ yieldsadiscreteframe
x x X
{ } ∈
ψ (see also [9]).
{ xi}i∈I
1
2000 Mathematics Subject Classification: 42C15, 46C99, 94A12, 46B25, 47A05.
Key words: Bonded operator, Hilbert space, continuousframe, atomic resolution of identity.
1
2 A.Fattahi, H. Javanshiri
In the recent years, there has been shown considerable interest by harmonic and
functional analysts in the frame of subspace problem of the separable Hilbert space;
see[5],[4],[3]and[2]andthereferencesthere. Frameofsubspacewasfirstintroduced
by P. Casazza and G. Kutyniok in [5]. They present a reconstruction formula f =
2 1
ν S π (f)forframesofsubspace. Continuousframeofsubspaceisanatural
i I i − Wi
∈
generalization from discrete frame of subspace to continuous.
P
As we expect, in discrete frameof subspaceevery element in has an expansion
H
in terms of frames. But in the continuous case it respect to Bochner integral which
is not desirable. Therefore, discretization of continuous frame of subspace is also
very important.
Suppose that the measure µ, which appears in the integral of continuous frame,
is Radon or discontinuous (Note that there exist infinite many positive finite dis-
continuous measure on a locally compact space X which are not counting measure).
Then x X : µ( x = 0 is nonempty set and we may investigate about some
{ ∈ { } 6 }
conditions under which every fixed element f has a countable subfamily J
f
∈ H
of X with frame property for h. This leads us to define uca-resolution of identity
(Definition 2.1), whichis ageneralization of theresolution of identity ([5], Definition
3.24), andatomic resolution ofidentity ([4]), toarbitraryHilbertspace(separableor
nonseparable). We then show that in this concept many basic properties of discrete
state can be derived within this more general context. In fact uca-resolution iden-
tity helps us to investigate continuous frames which have discretization. Because
under some extra conditions, every uca-resolution of identity provides a continuous
frame of subspace, and conversely. This means that the relationship between uca-
resolution of identity and known continuous frames, such as frame of subspace is
very tight.
Assume to be a Hilbert space and X be a locally compact Hausdorff space
H
endowed with a positive Radon or discontinuous measure µ. Let = W be
x x X
W { } ∈
a family of closed subspaces in and let ω : X [0, ) be a measurable mapping
H → ∞
such that ω = 0 almost everywhere (a.e.). We say that = (W ,ω(x)) is a
ω x x X
6 W { } ∈
continuous frame of subspace for , if;
H
(a) the mapping x π is weakly measurable;
7→ Wx
(b) there exist constants 0 < A,B < such that
∞
2 2 2 2
A f ω(x) π (f) dµ(x) B f (1)
k k ≤ X k Wx k ≤ k k
Z
for all f . The numbers A and B are called the continuous frame of subspace
∈ H
bounds. If satisfies only the upper inequality in (1), then we say that it is a
ω
W
continuous Bessel frame of subspace with bound B. Note that if X is a countable
Discretization of continuousframe 3
set and µ is the counting measure, then we obtain the usualdefinition of a (discrete)
frame of subspace.
For each continuous Bessel frame of subspace = (W ,ω(x)) , if we
ω x x X
W 2 { } ∈
define the representation space associated with by L (X, , ) = ϕ : X
ω ω
W H W { →
ϕ is measurable,
H|
2 2
ϕ(x) W and ϕ(x) dµ(x) < , then L (X, , ) with the inner product
∈ x Xk k ∞} H Wω
given by
R
2
ϕ,ψ = ϕ(x),ψ(x) dµ(x), for all ϕ,ψ L (X, , ),
ω
h i h i ∈ H W
X
Z
2
is a Hilbert space. Also, the synthesis operator T : L (X, , ) is define
Wω H Wω → H
by
T (ϕ),f = ω(x) ϕ(x),f dµ(x),
h Wω i X h i
Z
2 2
forallϕ L (X, , )andf . ItsadjointoperatorisT : L (X, , );
ω ∗ ω
∈ H W ∈H ω H → H W
W
T (f) = ωπ (f). For more details see [2].
W∗ω Wω
Now, we give two immediate consequences from the above discussion. As the
first, we have the following characterization of continuous Bessel frame of subspace
in term of their synthesis operators as in discrete frame theory; see [3].
Theorem 1.1 A family is a continuous Bessel frame of subspace with Bessel
ω
W
fusion bound B for if and only if the synthesis operator T is a well-defined
H Wω
bounded operator and T √B.
k Wωk ≤
Also, by an argument similar to the proof of ([3], Theorem 2.6), we have a
characterization of continuous frame of subspace as follows;
Theorem 1.2 The following conditions are equivalent:
(a) =( W ,ω(x)) is a continuous frame of subspace for ;
ω x x X
W { } ∈ H 2
(b) The synthesis operator T is a bounded ,linear operator from L (X, , )
Wω H Wω
onto ;
H
(c) The analysis operator T is injective with closed range.
∗
ω
W
If is a continuous frame of subspace for with frame bounds A,B, then we
ω
W H
define the frame of subspace operator S for by
Wω Wω
S (f)= T T (f), f ,
Wω Wω W∗ω ∈ H
which is a positive, self-adjoint, invertible operator on with A Id S
H · H ≤ Wω ≤
B Id .
· H
4 A.Fattahi, H. Javanshiri
2 Main result
For instituting a relationship between discrete and continuous frame of subspace,
we generalize the concept of continuous frame and resolution of identity to arbitrary
Hilbert space . For this propose, we introduce the summation to noncountable
H
form. Let be a Hilbert space and T be a family of bounded operators on
x x X
H { } ∈
it. If now, set Γ, the collection of all finite subset of X, then Γ is a directed set
ordered under inclusion.
Let f be a fixed element of the Hilbert space . Define the sum S(f) of the
H
family T (f) as the limit
x x X
{ } ∈
S(f)= T (f)= lim T (f): γ Γ .
x x
{ ∈ }
x X x γ
X∈ X∈
If this limit exists, we say that the family T (f) is unconditionally summable.
x x X
{ } ∈
It is easy to see that the family T (f) is unconditionally summableif and only
x x X
{ } ∈
if for each ε > 0, there exist a finite subset γ0 Γ such that
∈
T (f) T (f) < ε,
x x
k − k
x γ1 x γ2
X∈ X∈
for each γ1,γ2 > γ0. Therefore for each ε > 0, there is a finite subset γ0 of X such
that
T (f) < ε
x
k k
for all x X γ0. Hence for a fixed element f , if Tx(f) x X is unconditionally
∈ \ ∈ H { } ∈
summable, then J = x X : T (f)= 0 is countable.
f x
{ ∈ 6 }
Definition 2.1 Let be a Hilbert space and let ω : X [0, ) be a measurable
H → ∞
mapping such that ω = 0 almost everywhere. We say that a family of bounded opera-
6
tor T on is an unconditional continuous atomic resolution (uca-resolution)
x x X
{ } ∈ H
of the identity with respect to ω for , if there exist positive real numbers C and D
H
such that for all f ,
∈ H
(a) the mapping x T is weakly measurable;
x
7→
2 2 2 2
(b) C f ω(x) T (f) dµ(x) D f ;
k k ≤ X k x k ≤ k k
(c) f = T (f).
x XR x
∈
P
TheoptimalvaluesofCandDarecalledtheuca-resolutionoftheidentitybounds. It
followsfromthedefinitionandtheuniformboundednessprinciplethatsup T
x X x x X
∈ k k ∈
< .
∞
Discretization of continuousframe 5
Remark 2.2 (a) If f satisfies in (c), then as we mention in above, there is a
∈ H
countable measurable subset J (depends of f) of X such that
f
T (f)= 0,
x
for all x X J . So
f
∈ \
2 2 2 2
ω(x) T (f) dµ(x) = ω(j) T (f) µ( j )
x j
X k k k k { }
Z jX∈Jf
and condition (b) transform to
2 2 2 2
C f ω(j) T (f) µ( j ) D f
j
k k ≤ k k { } ≤ k k
jX∈Jf
(b) If is a separable Hilbert space with an orthonormal bases e , then by
n ∞n=1
H { }
condition (c), for each n there exists a countable measurable subset J of X such
n
that
T (e ) = 0,
x n
for all x X J . So, we can find a countable subset J = J of X such that
n ∞n=1 n
∈ \
S
T (f)= 0,
x
for all f and x X J, and we have
∈ H ∈ \
2 2 2 2
ω(x) T (f) dµ(x) = ω(j) T (f) µ( j ).
x j
X k k k k { }
Z j J
X∈
Therefore, if is a separable Hilbert space, Definition 2.1 and Definition 3.1 in [4]
H
coincide.
From now on is a Hilbert space with orthonormal bases eλ λ Λ and X is a
H { } ∈
locally compact Hausdorff space endowed with a positive Radon or discontinuous
measure µ, and ω : X [0, ) is a measurable mapping such that ω = 0 almost
→ ∞ 6
everywhere. For a fix element f , by [7] there exists a countable subset J of Λ
∈ H
such that f,e = 0 for all λ Λ J.
λ
h i ∈ \
The following is an important example of uca-resolution compatible with defini-
tion 2.1, and note that this example does not satisfy in the definition of resolution
of identity and atomic resolution of identity which stated in [5] and [4], respectively.
Example 2.3 Let be a Hilbert space with an orthonormal basis eλ λ Λ. If, we
H { } ∈
consider Λ as a locally compact space with discrete topology and measurable space
6 A.Fattahi, H. Javanshiri
endowed with counting measure, then the family Tλ λ Λ of bounded operators on
{ } ∈
, defined by
H
T (f)= e ,f e , for all f and λ Λ,
λ λ λ
h i ∈ H ∈
is an uca-resolution of identity for .
H
Inthenexttheoremweshowthateveryuca-resolution ofidentity for ,provides
H
a continuous frame of subspace.
Theorem 2.4 Let T be a family of bounded operators on and for each
x x X
{ } ∈ H
x X, set W = T ( ). Suppose that there exists D > 0 and R > 0 such that the
x x
∈ H
following conditions holds:
2
(a) f = ω(x) T (f)µ( x );
x X x
∈2 { 2} 2
(b) ω(x) π (f) T (f) dµ(x) R f ;
X P k Wx − x k ≤ k k
2 2 2
(c) ω(x) T (f) d µ(x) D f ,
RX k x k ≤ k k
for all f . Then (W ,ω(x)) is a continuous frame of subspace for .
R x x X
∈ H { } ∈ H
Proof. Let f be a fix element of . as we mention in remark 2.2(a), there exists a
H
countable subset J of X such that
f
2
ω(x) T (f)µ( x ) = 0,
x
{ }
for all x X J , and
f
∈ \
2 2 2 2
ω(x) T (f) dµ(x) = ω(x) T (f) µ( x ).
x x
k k k k { }
X
Z x X
X∈
So we can use Cauchy-Schwarz inequality and compute as follows
4 2 2
f = ( f, ω(x) T (f)µ( x ) )
x
k k h { } i
x X
X∈
2
= ( ω(x) µ( x ) f,ω(x) µ( x ) T (f) )
x
h { } { } i
xX∈X q q
2
= ( ω(x) µ( x ) π (f),ω(x) µ( x ) T (f) )
h { } Wx { } x i
xX∈X q q
2
( ω(x) µ( x ) π (f) ω(x) µ( x ) T (f) )
≤ k { } Wx kk { } x k
xX∈X q q
2 2 2
( ω(x) π (f) µ( x ))( ω(x) µ( x ) T (f) )
≤ k Wx k { } k { } x k
xX∈X xX∈X q
2 2 2 2
( ω(x) π (f) dµ(x))( ω(x) T (f) dµ(x))
≤ k Wx k k x k
x X X
Z ∈ Z
2 2 2
D f ( ω(x) π (f) dµ(x)).
≤ k k k Wx k
X
Z
Discretization of continuousframe 7
Also, by triangle inequality and hypothesis we have
R
2 2 2 2
ω(x) π (f) dµ(x) D(1+ ) f ,
X k Wx k ≤ sD k k
Z
so the assertion holds. 2
Casazza and Kutyniok in [5] introduced an interesting example of atomic resolu-
tion of identity. In the next theorem we obtain the uca-resolution of identity form,
which is the converse of theorem 2.4.
Theorem 2.5 Let (W ,ω(x)) be a continuous Bessel frame of subspace for
x x X
{ } ∈
with Bessel bound D, and for each x X, let T : W be a bounded operator
x x
H ∈ H →
such that T π = T . Also assume that for each f
x Wx x ∈ H
2
f = ω(x) T (f)µ( x ).
x
{ }
x X
X∈
Then for all f we have
∈ H
1
2 2 2 2
f ω(x) T (f) dµ(x) DE f ,
x
Dk k ≤ k k ≤ k k
X
Z
where E = sup T .
x X x x X
∈ k k ∈
Proof. By similar prove of Theorem 2.4, we obtain
1
2 2 2
f ω(x) T (f) dµ(x).
x
Dk k ≤ X k k
Z
Also we have
1
2 2 2
f ω(x) T (f) dµ(x)
x
Dk k ≤ k k
X
Z
2 2
= ω(x) T π (f) dµ(x)
k x Wx k
X
Z
2 2 2
ω(x) T π (f) dµ(x)
≤ k xk k Wx k
X
Z
2 2 2
E ω(x) π (f) dµ(x) DE f .
≤ k Wx k ≤ k k
X
Z
Whence, for each f
∈ H
1
2 2 2 2
f ω(x) T (f) dµ(x) DE f .
x
Dk k ≤ X k k ≤ k k
Z
as we required. 2
8 A.Fattahi, H. Javanshiri
Proposition 2.6 Let W be a family of closed subspace of Hilbert space
x x X
{ } ∈ H
such that the mapping x π is weakly measurable. Also suppose ω is a bounded
→ Wx
map and the following conditions hold for all f :
∈ H
(a) There exists C > 0 such that
1
2 2
π (f) dµ(x) f ,
X k Wx k ≤ C k k
Z
(b) f = ω(x)π (f)µ( x ).
x∈X Wx { }
Then (W ,ω(x)) is a continuous frame of subspace for .
{ Px }x∈X H
Proof. By condition (a) we see that
sup ω(x)
ω(x)2 π (f) 2 dµ(x) x X f 2, (f ).
X k Wx k ≤ ∈C k k ∈ H
Z
Condition (b) implies that for a fixed element f of
H
2 2 2 2
ω(x) π (f) dµ(x) = ω(x) π (f) µ( x ),
X k Wx k k Wx k { }
Z x X
X∈
and
2 2
π (f) dµ(x) = π (f) µ( x ).
k Wx k k Wx k { }
X
Z x X
X∈
Now,sincethefamily ω(x)µ( x )T isunconditionalsummable,wecanuseCauchy-
x
{ { } }
Schwarz inequality and compute as follows
4 2
f = ( ω(x)µ( x )π (f),f )
k k h { } Wx i
x X
X∈
2 2
= ( ω(x)µ( x ) π (f) )
{ } k Wx k
x X
X∈
2 2 2
( ω(x) µ( x ) π (f) )( π (f) µ( x ))
≤ { } k Wx k k Wx k { }
x X x X
X∈ X∈
1
2 2 2
f ( ω(x) π (f) µ( x ))
≤ Ck k k Wx k { }
x X
X∈
Thus
2 2 2 2 2
C f ω(x) π (f) µ( x ) = ω(x) π (f) dµ(x)
k k ≤ k Wx k { } X k Wx k
x X Z
X∈
for all f , and this complete the proof. 2
∈ H
In the following proposition we give a reconstruction formula for continuous
frame of subspace in the special case.
Discretization of continuousframe 9
Proposition 2.7 Let W be a family of orthogonal closed subspace of Hilbert
x x X
{ } ∈
space . If (W ,ω(x) is a continuous frame of subspace for with bounds
x x X
H { } ∈ H
C,D, then for each f
∈ H
f = π (f).
Wx
x X
X∈
The converse is true if ω is bounded and there exists C > 0 such that
1
2 2
π (f) dµ(x) f ,
k Wx k ≤ Ck k
X
Z
for all f .
∈ H
Proof. Let (W ,ω(x) be a continuous frame of subspace. First, we should
x x X
{ } ∈
noted that for each f , by Hahn-Banach Theorem and orthogonality of the
∈ H
family W , there exists a sequence f in such that f f and for each
x x X n n
{ } ∈ { } H −→
n we have the following equality
f = π (f ).
n Wx n
x X
X∈
Now we define S (f) = π (f), where γ is an arbitrary finite subset of X
γ x γ Wx
∈
and f . Therefore
∈ H P
2 2 2
C S (f) f ω(x) π (S (f) f) dµ(x)
k γ − k ≤ k Wx γ − k
X
Z
2 2
ω(x) π (f) dµ(x)
≤ k Wx k
X
Z
2
D f .
≤ k k
By replacing f with f f we obtain
n
−
D
S (f f) (f f) f f .
γ n n n
k − − − k≤ sCk − k
The converse holds by 2.6. 2
Now we want to show that, by a given uca-resolution of identity, each f has
∈ H
a new countable reconstruction formula. First we need the following Lemma:
Lemma 2.8 Let T be an uca-resolution of the identity with respect to weight
x x X
{ } ∈
ω for with bounds C and D, and let f be a frame sequence. Then there
i i I
H { }∈
exists a countable subset J of X, such that ω(j) µ( j )T (f ) is a frame
{ { } j∗ i }i∈I,j∈J
for span f .
i i I p
{ }∈
10 A.Fattahi, H. Javanshiri
Proof. If we set J = x X : T (f ) = 0 , then by definition of uca-resolution
i x i
{ ∈ 6 }
of identity , J is a countable and measurable subset of X. Now, set J = J .
i i I i
∈
So J is a countable and measurable subset of X, and for each f span f and
∈ { i}Si∈I
x X J we have
∈ \
T (f)= 0.
x
Hence we see that for each f span f
i i I
∈ { }∈
2 2 2 2
C f ω (j)µ( j ) T (f) D f ,
j
k k ≤ { } k k ≤ k k
j J
X∈
and
f = T (f),
j
j J
X∈
and these series converge unconditionally.
Now,supposethatAandB areframeboundsof f . Foreachf span f
i i I i i I
{ }∈ ∈ { }∈
we have
2 2 2 2
A ω (j)µ( j ) T (f) < ω (j)µ( j )T (f),f >
j j i
{ } k k ≤ | { } |
j J j J i I
X∈ X∈ X∈
2 2
B ω (j)µ( j ) T (f) ,
j
≤ { } k k
j J
X∈
and therefore
2 2 2
AC f A ω (j)µ( j ) T (f)
j
k k ≤ { } k k
j J
X∈
2 2
< f,ω (j)µ( j )T (f ) >
j∗ i
≤ | { } |
j J i I
X∈ X∈
2 2 2
B ω (j)µ( j ) T (f) BD f .
j
≤ { } k k ≤ k k
j J
X∈
and this complete the proof. 2
Theorem 2.9 Let T be an uca-resolution of the identity with respect to
x x X
{ } ∈
weight ω for with bounds C and D. Then for each f , there exists a countable
H ∈ H
subset I (dependents on f) of X, such that we have the following reconstruction
formula
2 1 2 1
f = ω (i)µ( i )S T T (f) = ω (i)µ( i )T T S (f),
− i∗ i i∗ i −
{ } { }
i I i I
X∈ X∈
where S is a frame operator of a frame sequence.
