Frames and Riesz Bases - Symmetry of Concepts
 
            Hans G. Feichtinger, Nov. 1995 

pamphlet: somehow influenced through Amons Ron's consideration of
  non-fundamental frames, i.e. frames for subgroups, and the work
  of Ole Christensen and Chris Heil on perturbation of frames. 


We have orthogonal bases, for the full Hilbert space, or just for
a (closed) linear subspace of a Hilbert space (non-complete ONS),
as the most "beautiful" systems for signal expansions (orthogonal
expansions are trivial, however it might be tricky to find a good
orthogonal system with certain properties, such as a wavelet basis).

There are also Riesz bases (no-more orthogonality, but with some
biorthogonal Riesz basis for the SAME space), somehow "distorted"
orthonormal bases (by some invertible, but not unitary mapping on
the Hilbert space), and of course Riesz-basis for a subspace.
It is easier to obtain Riesz bases (e.g. one even can have symmetric
wavelets, forming a Riesz basis, but one cannot have them orthogonal),
and as soon as the biorthogonal basis has been determined they are -
from a practical point of view, computationally at the some level
as orthogonal systems.

Somehow - in the usual ranking of systems - frames come next, or what
Amon Ron calls "fundamental frames". Systems, which my be "overcomplete",
but allow expansions of arbitrary elements of a Hilbert space in the form
of a series with (non-unique, but) l2-coefficients. It is also well-known
fact, that the minimal norm coefficients (the solution of the
MNLSQ-problem, the minimal-norm least-square solution, which actually
is the same as the Moore-Penrose inverse) are obtained by means of the
so-called dual frame. And, as a next step prepared by our explanations,
one can have what I would like to call LOCAL frames, i.e. frames for
linear subspaces of the Hilbert space, and since one has only one closed
subspace which can be used for this purpose it is equivalent to the
statement that elements in the closure of the finite linear combinations
of the "frame"-elements can also be represented by series with  
l2-coefficients.

There are two - also well known - characterizations of both Riesz bases
(for a subspace) and (fundamental) frames: and they look alike...
.. the purpose of these comments is to make this more explicit and 
comment on variations of this principle.

In terms of linear algebra (to pin-point the linear background of the
story, or to have a simple-minded model) a MATRIX can be used in two
ways, or in other words, a matrix-vector multiplication can be read
in two ways. For simplicity we take the usual matrix-on-vector 
multiplication:  y = A * x (in MATLAB symbols), where  y,x are
real column vectors of size  N and K respectively, and A  is some 
real matrix of size  K x N. 

We may now interpret at a LINEAR combination of columns of M, with
coefficients given by the coordinates of the vector x,  OR , the
other way round, as the collection of scalar products of all the
N row vectors (building up the matrix A) with the fixed vector x.
In the first case, we read  matrix multiplication  A  as a mapping
from some coefficient space into a space of vectors, the other way
we read it as a mapping from the coefficient space to the vector space.
[Sometimes people like to do matrix multiplication from the left or
the right, but this does not appear to be necessary for our explanation,
maybe even counter-productive]. 

The basic notions of linear algebra are related to these two
interpretations, and the basic questions about injectivity and
surjectivity of mappings between (possible different) sets:
The rows of A are linear independent (by definition) if the mapping
x >> A *x  is injective.
The rows of A  SPAN the Euclidean N if the mapping
x >> A * X is surjective. 
[Of course there is a lot more to be said here, e.g. in connection
with adjoint mappings, transposition of matrices, etc.., kernels
on non-injective linear mappings, but we have to skip this here]. 
Only the fact  that the null-space of a linear mapping is exactly
the same as the range of the adjoint mapping, given by the transpose
(and conjugate) matrix A*, is mentioned here. 

Now think of the rows of M (or of the colums of M) as systems
of vectors (of the right dimension), then it is clear that similar 
notions will be available for Hilbert spaces in general. It is here,
where Riesz bases (defining injective, but maybe not surjective) 
mappings from a coefficient space into the Hilbert spaces correspond
to our first interpretation, and frames (surjective mappings, which 
do not have to be injective, but define injective mappings from the
Hilbert space in the the coefficient space) correspond to our second
interpretation of the role of "systems of vectors" (in a Hilbert space).

Now, for infinite dimensional Hilbert spaces  (i.e. for spaces which are 
too big to allow any finite basis) one has to be somewhat more careful
(in order to preserve facts which are "trivial" for finite dimensional
vector spaces), and make distinctions between the linear span of a set
of vectors and the closed linear span (the closure of the set of all
ordinary linear cominations with an arbitrary - but finite - number 
of terms in the sense of Hilbert space convergence).

As can be shown by functional analytic arguments (cf. e.g. Ole
Christensen's thesis) the right characterization (equivalent to the    
ordinary frame condition

  A|f|^2 <=  sum  |<f,f_i>|^2   <=  B|f|^2  for all f in H 

is the statement, that the COEFFICIENT mapping  f >>  <f,f_i> 
defines a bounded, linear and injective mapping from the Hilbert
space into the sequence space  l2(I) (over the index set I). 

On the other hand a family (f_i)  defines a Riesz basis (for 
a subspace of a Hilbert space H) if and only if one has 
(by defintion)

  A|c|  <=    | sum  c_i f_i |^2  <= B |c|   for all  c in l2(I) 

(in order to stress the similarity of formulations). Again, this
synthesis mapping  c >> sum c_i f_i  is bounded and injective
(and obviously has closed range in H).  

In both cases one can get extra information out of the defintion
as it stands (and Charly Groechenig stresses that they might not
come automatically if one goes to analog definitions in the case
of Banach spaces):

For a given frame (according to the defintion above) there is also 
the adjoint frame operator, which is a bounded mapping back from
the coefficient space to the Hilbert space, which indeed is an
invertible operator (due to the injectivity of the frame coefficient
mapping). Furthermore there is some left inverse to the compose
(coefficient mapping followed by synthesis, this operator is usually
called the FRAME operator), and as a further consequence the range
of the coefficient mapping is closed (the synthesis mapping followed
by the coefficient mapping - reversed order now - gives the projection
operator). 

Of course, in the case of the Riesz basis, we see that the projection
of the Hilbert space is just the coefficient mapping followed by the
synthesis mapping (since the coefficients with respect to a Riesz
basis are uniquely determined, and the orthogonal complment of the
closed linear space generated by the Riesz basis can be neglected 
anyway. 

Let us - for the sake of unification of terminology  recall a notion,
which plays an important (technical) role in interpolation theory, 
where it is all to often used in order to transfer interpolation
results (with respect to complex or real interpolation methods) 
known to be true for vector-valued Lp spaces to more general Banach spaces.
(this principle has been heavily used by myself and coauthors for
the interpolation of Wiener amalgam spaces, later modulation spaces,
or coorbit spaces..). Certainly people working in geometry of Banach
spaces know a lot about further occurrences of this concept.  

Definition:   A (think "smaller")  Banach space  B1  is called a 
RETRACT  of a (think "larger") Banach space  B2, if the following
is true:  
B1 can be identified with a closed and complemented subspace of B2.

Mathematically (and practically) this means: there is a linear and
injective mapping  C  from the Banach space  B1 with closed range, and
a projection mapping from the (big) space P  B2 onto the closed range
C(B1) in B2, which is equivalent to the following description:

There exists a continuous, linear mapping  C  from B1 into B2,
and some other continuous, linear mapping  R  from B2 into B1,
such that   R C = Id_B1   (C R is just P as discussed above). 

It is clear to see that ALL the information (about the relationship
between  B1 and B2) are described by the mapping C (its range is
naturally isomorphic to B1, but also a closed subspace of B2, thus
both norms - the inherited B2-norm and the B1-norm, obtained by
"transport of norms" through the injective mapping C  are equivalent
on C(B1)).  Exactly this equivalence of norms is what makes the
"similarity" of the above two formulas (definition of frames and
Riesz bases).  

So the only difference between the two notions (both describe
a retract, defined by means of a system of elements in some 
Hilbert space) is in the DIRECTION in which the mapping goes
(from the "abstract" Hilbert space to the "standard" and familiar
l2-space, or in the opposite direction), and thus the question
whether the system is understood in the first or second sense
(as described in the very first part, in terms of linear algebra). 

We can also summarize by saying, that in the frame case, the
Hilbert space is the "small" one, and l2(I) is the big one,
and conversely, in the case of the Riesz basis, the Hilbert
space is the small one.

It is a good exercise opening up your mind for the situation
given here to think of the Hilbert space H itself of l2 itself.
Then the question, which one of the notions (frame or Riesz basis
for a subspace) is "better", or corresponds to the stronger concept,
turns out to be "non-sense". Also, this symmetric point of view 
might help to transfer results in the literature about Riesz 
bases to frames (and maybe even in the other direction). 

A phenomenon becomes interesting (this is now really infinite
dimensional): We can talk about frames for closed subspaces
of some Hilbert space. In that case we have a symmetric situation,
each spaces is identified with a closed subspace of the other,
without forcing the two spaces to be "naturally" identified.
Indeed, if we have a (non-fundamental) frame than we have a
bounded, linear mapping from H1 into a closed linear subspace
of the Hilbert space H2 = l2(I), but also another mapping (its
adjoint mapping) back.

It would thus be natural to call in this situation of a 
DOUBLE RETRACT (between  H1 and H2), maybe some better terminolgy?? 



This situation can be seen as VERY similar (also this analogy
can be worked out in detail) to the situation one is looking at
in the case of the SVD (singular value decomposition).  

The standard picture is the following: 
Let us talk about a linear mapping from R3 into R3 of rank 2,
i.e. a  3x3 matrix  A  which contains two (but not three) linear
column (or row) vectors, and thus defines a linear mapping with
a nontrivial and one-dimensional kernel  null(A). The adjoint
mapping (defined by the transpose conjugate matrix A*)  then
has also a two-dimesional ( 3 - dim(null(A))) range, which is
actually the orthogonal complement of null(A), and at the same
time the range of  A  range(A) coincides with the orthogonal
complement of null(A*) in a very natural way. 

The combined mappings (now given by symmetric matrices)  AA* and A*A
map now  (one copy of) R3 into the other, and actually are projections
onto the range of  A or A* respectively (the one which comes last).
[SVD tells us further that one can choose orthonormal systems of
eigenvectors of AA* and A*A respectively, which are mapped through
A and A* respectively onto each others]. For now we only need the
fact that the two ranges  range(A) and range(A*) are isomorphic
(through  A* and A respectively) to each other, and closed subspaces
in the space to which they belong (trivial in this case because we 
only have finite dimensional spaces, but for compact operators on
Hilbert spaces the situation is exactly the same, but with
infinite-dimensional, isomorphic and closed subspaces). 

From this point of view it is NOT surprising that there are
many links between frame theory and the discussion of SVD and
the Moore-Penrose inverse. Indeed, if we take the situation
described in the last paragraph, then we see that the linear 
mapping  A from R3 into R3  (if we read it as the trivial 
projection from R3 onto range(A*), followed by the identification
mapping - induced by A - between range(A*) and range(A))  has
a NATURAL symmetric companion, the mapping  A+ (pseudo-inverse
matrix, or Moore-Penrose inverse), namely the mapping which 
first projects  R3 (on the range(A) side) down to range(A) and
than maps range(A) back (isomorphically) to  range(A*), a closed
subspace in the domain of the original mapping A.
[Suggestion: draw a sketch of the geometric situation].

It is exactly this kind of situation which we have to look
at if we have 'double retracts', with the only EXCEPTION, that
NOW we do not have anymore the natural IDENTIFICATION  of
the two subspaces ( range(A) and range(A*) above) by any
natural mapping. 

  
It is now obvious how to define a double retract properties
between a pair of Banach spaces.

Let us go back to the concept of frames, understood as
a retract between some Hilbert space in connection to l2(I).

As we have seen in full generality in our atomic theory
based on integrable group representation (the analogy of
this situation to the case of PSI = principal shift invariant 
or spline-type spaces will be discussed in more detail elsewhere),
it is of great interest to EXTEND this point of view (embedding
of coorbit spaces into sequence space) from the ISOLATED 
point of view of the Hilbert space L2 to a rich FAMILY of
BANACH SPACES (of a similar form). Recall that the wavelet
transform (maybe sampled sufficiently dense) extends to
a retract between Besov-Triebel-Lizorkin spaces to weighted 
mixed-norm spaces of a certain type, while modulation spaces
are strongly tied - by means of the Short-Time-Fourier Transform - 
to very similar weighted mixed-norm spaces over Z^2. 

Thus (to come back to the one-sided version of the story,
but now emphasizing the richness of families of "alike"
Banach spaces) it appears to be natural to us to suggest
the concept of a  BANACH FAMILY FRAME:
(a frame for a family of Banach spaces, NOT a family of
individual Banach frames, is one of the aspect of our concept):

Given two compatible families of Banach spaces  BB1 and BB2 ,
with a natural identification through certain parameters, 
(cf. the book by Bergh-Loefstroem on interpolation for this 
notion), we call a mapping  C: BB1 to BB2  a BFF 
if the following conditions hold true:    

C, restricted to any of the members B1 of  BB1 maps B1
injectively onto a closed subspace of the "corresponding"
(through the parameters) Banach space B2 out of BB2.  

Note that it MAKES SENSE to talk of ONE mapping C between
such families, because whenever one restricts  C  to  two
spaces B1 and C1 out of BB1 the restriction to their intersection
will be the same. Recall, that the definition of the 
Fourier transform on Lp-spaces (e.g. for 1 <= p <= 2) 
is understood in exactly the same way: On L1 intersected L2
"the Fourier transform" is defined in the same way, giving
exactly the some f^, if it is defined as the classical FT
using integrals, or the Plancherel approach (which would be
the limit of the finite integrals from -n to n).  
 
There is more to say about this terminology, but let me finish
for now. Let me just say, that much of the effort connected
with our atomic decomposition results, or the irregular sampling
problem, is related to the question of the following type:
WHEN can a given frame (in the Hilbert space sense) be extended
to a  BANACH FAMILY FRAME, and WHAT are the corresponding spaces.
Indeed, starting from L2-frame theory based on a given integrable  
representation have to DEFINE (the coorbit construction),
from rather general translation invariant Banach spaces Y  on
the locally compact group G which defines this representation 
BOTH the Banach space  Co(Y) (the coorbit space)  AND the
Banach space of sequences Y_d  (a discrete version of Y),
in order to show THEN that the (continuous-selective) description
matches well the (discrete-constructive atomic) description
bases on Y_d, and that we are in exactly this situation
of a BANACH FAMILY FRAME between the family of coorbit spaces
and the corresponding sequence spaces Y_d (both of them carrying
the original Banach spaces Y as "parameter" for the purpose
of matching of parameters, or in order to "see" which one
is the "corresponding" space on the other side).  

In THAT situation it is even an extra interesting aspect that
EACH sufficiently dense and well separated family of points,
and each sufficiently "good" atom gives raise to such a 
BFF and that all properties (bounds) can be obtained in
a unified way (e.g. bounds depend only on the degree of
density, and NOT on the particular family of points under
discussion).  

Final comment: it is of course possible, to have a variety
of retract or double retract mappings between any given pair
of Banach spaces, so on most cases one should not just say
that "the are related through some double retract property",
but rather emphase, that a CONCRETE pair of linear mapping
induced the situation of double retracts.

Arguments of this form have been used heavily in the second
part of our atomic decomposition theory (Fei-Groch,
Monatshefte, for coorbits in connection with the corresponding
Banach spaces of sequences) , in order to verify that Banach
space properties of ONE space (e.g. reflexivity) goes over to
the other one (the coorbit space, in our case).