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\begin{document}
\leftline{ \scriptsize \it SPb MatOb Preprints,
(2006), pp, 1-6.}
\vspace{1.3cm}
\title
{Measurable space frames revised
}
{\thanks{\enskip This research was done with partial support
by the Fond RFFI Grant 06-01-00457
}}
{\thanks{\enskip This is a revised version of Preprint SPbMO 2005-9
}}
{\thanks{\enskip \copyright 2006 Oleg I. Reinov, SPbGU, S. Petersburg
}}
\maketitle
\date{}
\begin{center}\author{O.~I. Reinov\\
\small{ Oleg Reinov, \\ Saint Petersburg State University,
Saint Petersburg, Russia,\\
orein@or1146.spb.edu
}}\end{center}
\begin{abstract}
A new notion of "Measurable space frame" is introduced;
the simplest but important properties are given. The next
parts of the work will be given in next papers.
\vskip 0.4 true cm
\noindent
\noindent
{\it Key words } : Frame, measurable, Hilbert space.\\
\end{abstract}
\maketitle
\pagestyle{myheadings} \markboth{\centerline {\scriptsize Oleg Reinov
}}
{\centerline {\scriptsize %Title
Measurable space frames
}}
%%%%%%%%%% the following introduction form is an option %%%%%%%%%%%%
\bigskip
\bigskip
\medskip
\begin{center}
%\section{\textbf{Heading of Section}}
\section{\textbf{Introduction}}
\end{center}
Let $H$ be a separable Hilbert space (i.e.
one can find in $H$ a dense countable system of vectors).
Let $\{e_j\}_{j=1}^{N},$ where $N\in \Bbb N\cup \{+\infty\},$
be a sequence of orthonormal elements in $H.$
Then, for any element $e\in H,$ we have
$$
\sum_{j=1}^{N} |\langle e, e_j\rangle|^2\le ||e||^2. \eqno{(1.1)}
$$
This is Bessel's inequality.
For {\it complete} orthonormal sequences
$\{e_j\}_{j=1}^{N}$ (that is, for orthonormal bases in $H)$
one has a stronger property of the systems:
$$
\sum_{j=1}^{N} |\langle e, e_j\rangle|^2= ||e||^2\quad
\text{ for every } \ e\in H. \eqno{(1.2)}
$$
This is Parseval's equality.
The relations (1.1) and (1.2) are, surely, most of the main ones
in the classical theory of Hilbert spaces. But we can go further,
considering countable or uncountable infinite (orthogonal or not)
families
in Hilbert spaces, even not necessarily separable, and trying
to define families for which somethings like (1.1) and (1.2) are
fulfilled. In this way, to be short, we can get naturally an
"overfilled"
(in most cases, nonorthogonal) families
which are very similar to Hilbert bases in separable (or nonseparable)
spaces --- in the sense that they may have the main properties almost like
the properties of the sequences in (1.1) and (1.2). If such
"overfilled" uncountable families are considered in
separable $ H,$ then clearly
their elements do not give us usual Hilbert bases since each
such family is uncountable and so is linear dependent.
Again to be short, let us say that we are on the way to obtain
a new class of orthogonal families, not bases but showing
themselves like bases and, so, with properties which can be
very useful in Hilbert space theory. These are {\it frames.}
More precisely, let $ \{ e_\al\}_{\al\in \Cal A}$ be a family
in $ H$ (we suppose usually that all te sets under consideraton
are not empty, so is $ \Cal A).$
One says that the family $ \{ e_\al\}_{\al\in \Cal A}$
is a {\it frame}\ if there exist two constants $ A$ and $ B,$
where $ 0< A\le B<\infty,$
so that for each $ e\in H$ the following relations are fulfilled:
$$
A\,||e||^2\le \sum_{\al\in\Cal A} |\|^2 \le B\,||e||^2 \eqno{(1.3)}
$$
(see [1, 2] for nice inroductions to the Frame theory).
If so, then $A$ is an {\it lower $($frame--$)$bound}\ and
$B$ is an {\it upper $($frame--$)$bound}\ for this frame.
If a family $E:= \{ e_\al\}_{\al\in \Cal A}$
satisfies a condition analogues to (1.1), then they say about
{\it Bessel family}\ and {\it Bessel frame}\ if a frame.
If the family $ \{ e_\al\}_{\al\in \Cal A}$
satisfies a condition analogues to (1.2), then they say about
{\it Parseval family}\ and in this case $ \{ e_\al\}_{\al\in \Cal A}$
is necessarily a frame, so we get {\it Parseval frame.}
More precisely, $ E$ is said to be a {\it Bessel family}
(respectively, a {\it Bessel sequence}) if there exists $ B>0$
such that for every $ e\in H$
$$ \sum_{\al\in\Cal A} |\|^2 \le B\,||e||^2
$$
(respectively, and the index set $ \Cal A$ is countable).
If $ E$ is a frame with (1.3), then it is said to be
a {\it Parseval frame} if in (1.3) $ A=B=1.$
More general, if $ A=B>0$ in (1.3) then the frame $E $
is called a {\it tight frame.} Two more definitions:\,
a frame $ E$ is {\it uniform,}\ if $ ||e_\al||=||e_\beta||$ for
all $\al, \beta\in \Cal A;$\, a frame $ E$ is {\it exact,}\
if the family $ E\setminus \{ e_\gamma\}$ is already not frame
whatever $ \gamma\in\Cal A$ to be.
Frame theory (cf. [1, 3]) has been developed remarkably during the last
about 30 years. We would not like to consisider even first
(but rather interesting and sometimes non-trivial) results of
the theory. Almost just now we will mention some ways
to generalize the notion of frame (and then we will study
one of them).
In which directions one can generalize Frame notion?
\small
$\Bbb A)$\
One way is to replace each $ e_\al$ in(1.3) by a collection of vectors
$ \{ e_{\al,j}\}_j.$ How to do this? The scalar products in (1.3) are,
roughly speaking, (ortho)proiections from $ H$ onto the corresponding
one dimensional spaces. So, if we want to change the situation in the
desiered direction, it is naturaly to consider, for each $ \al,$
a subspace $ E_\al$ instead of one dimensional subspace
$ \operatorname{ span}\, \{ e_\al\} $ and to take the orthogonal
projecton $ \pi_{E_\al}$ insted of the "projection" $ \< \cdot,e_\al\>.$
This is the way P.~G. Casazza and G. Kutyniok [2] went.
Here is their corresponding definition.
Let $I$ be some index set, and let $\{v_i\}_{i\in I}$ be a family of weights,
i.e., $v_i > 0$ for all $i\in I.$ A family of closed subspaces
$\{W_i\}_{i\in I}$ of a Hilbert space $H$ is a {\it frame of subspaces
with respect to $\{v_i\}_{i\in I}$ for $H,$ }
if there exist constants $0 < C \le D <\infty$ such that
$$C\,||f||^2 \le \sum_{i\in I} v_i^2
||\pi_{W_i}(f)||^2\le D\,||f||^2 \ \text{ for all }\ f\in H.$$
The notions of bounds, upper and lower bounds,
tight, Parseval etc. frames are defined by a natural way.
\small
$\Bbb B)$\
Another way for a generalization. We iust change the sums in (1.3)
by integrals, what means that we just replace the index set $ \Cal A$
by a positive measure space, e.g. $(\Omega, \Sigma, \mu).$
At the moment, we do not discuss this possibility, since
essentially just now
we will give the exact definitions even for more general situation.
\med
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2d part
\begin{center}
\section{\textbf{"Integral" definition of Frame
}}
\end{center}
We introduce a new notion of "Measurable space frame" and begin the
investigation with its simplest properties.
Let us look at the condition (1.3) in the definition of frame
from another point of view.
Put $ \Omega:=\Cal A$ and let $ \mu_c$ be the "counting measure"
on $ \Cal A,$
i.e. $ \mu_c$ is the measure on $ \sigma$-algera of all subsets of the
set $ \Cal A$ with the property that for each point $ \al\in\Cal A$
one has $ \mu_c(\{\al\})=1.$ Note that for each nonnegative function
$ \ffi$ on $ \Cal A$ the following equality holds:
$$ \sum_{\al\in\Cal A} \ffi(\al) =\int_{\Cal A} \ffi(\al)\,d\mu_c(\al)
\equiv \int_{\Omega} \ffi\, d\mu_c.
$$
Thus, the relations (1.3) can be rewritten in an "integral form", and we get
that
the family $ \{ e_\al\}_{\al\in \Cal A}$
is a frame if there exist two constants $ A$ and $ B,$
where $ 0< A\le B<\infty,$
so that for each $ e\in H$ the following relations are fulfilled:
$$
A\,||e||^2\le \int_{\Cal A} |\|^2\,d\mu_c \le B\,||e||^2.
\eqno{(1.3a)}
$$
\med
%%%%%%%%%%%%%%%%%%%%%%%% 3d part
\begin{center}
\section{\textbf{Measurable space frame
}}
\end{center}
Combining this "integral idea" with ideas of
P.~G. Casazza and G. Kutyniok [2], we come to a natural generalization
in both directions (I \& II) of the notion of frames as well as the notion of
frame of subspaces.
Bellow, when considering "any--valued" functions on a set
with a (nonnegative) measure,
{\it we will always suppose that the functions are measurable;}
"positive" will mean "positive a.e." etc.
E.g., considering a composition of type
\small
$ \begin{array}{c}
||g(\cdot)||:\quad
\text{ SET (with a measure) }
\overset{g}\to%{\to} \phantom{AAAA}
\\ {}
\phantom{AAAAA}
\left\{\text{collection of linear subspaces
of a Hilbert space} \right\}
\overset{||\cdot||}\to%{\to}
\Bbb R \\
\end{array} $
\small
\noindent
we assume that, at least, the real--valued function $
||g||$ is measurable. What about the question when the desired
measure conditions are fulfilled, it is a theme of other separate
investigations.
For a lot of very useful information on vector integrations or
nonscalar--valued mesurable functions (which will be used,
in particular, in this work which is supposed to be continued in
the next papers), see a nice written book by J. Diestel and
J.~J. Uhl [4] (this is only the author's recomendation; the reader
can use also any corresponding textbook on the topic).
\med
\subsection*{\textbf{Definition}}
Let $(\Omega, \Sigma, \mu)$ be a measure space
and let $\ffi$ be a positive measurable function on $ \Omega.$
$1)$\
A family $\{f_{\omega,j}\}_{\omega\in\Omega,j\in J_\omega}$
of vectors of a Hilbert space $H$ is said to be an
{\it integral frame} (with respect to $\ffi)$ for $H$
if there exist some constants $0|^2\, d\mu\le
D\|f\|^2\ \ \forall\, f\in H. \eqno{(1.4)}
$$
More general,
$2)$\
A family
$ \(F_\omega\)_{\omega\in\Omega}$
of closed (linear) subspaces of a Hilbert space $ H$ is said to be {\it a
measurable space frame, MSF,}\ for $ H$ with respect to $ \ffi,$
if there exist some constants $00$ for every $ \omega\in\Omega$ \
and let
$ \left\{ f_{\omega\,j}\right\}_{j\in J_\omega}$ be
a frame in $ H$ with upper bounds $ B_\omega$ $($upper$)$ and
$ A_\omega$ $($lower$).$
For all $ \omega\in\Omega,$ let us put
$ F_\omega=\overline {span}_{j\in J_\omega} \left\{ f_{\omega\,j}\right\}$
Take, in each of the subspace
$ F_\omega$ an orthonormal basis
$ \{e_{\omega\,j}\}_{j\in J_\omega}.$
Suppose that
$ 0|^2\le \\
\int_{\Om} B_\om \ffi^2 \|\pi_{F_\om}(f)\|^2 \le
B\int_{\Om} \ffi^2 \|\pi_{F_\om}(f)\|^2. \\
\end{array}
$
\small
Note that
$$
\int_\Om \sum_{j\in J_\om} |\<\pi_{F_\om}(f), \ffi\,f_{\om j}\>|^2=
\int_{\Om} \sum_{j\in J_\om} | < f, \ffi(\om)f_{\om j}>|^2.
$$
Therefore, if
$ \left\{ \ffi(\omega)\,f_{\omega\,j}\right\}_
{\omega\in\Omega\,j\in J_\om}$
is a frame for $ H$ with bounds $ C,D,$ then
$ \{F_\om\}$ is a MSF for $ H$ with bounds
$ C/B$ and $ D/A$. Moreover, if
$ \left\{ F_\om\right\}$ is a MSF with bounds
$ C,D,$ then it follows from above computations that
$ \left\{ \ffi(\omega)\,f_{\omega\,j}\right\}_
{\omega\in\Omega\,j\in J_\om}$
is an integral frame for $ H$ with bounds $ AC, BD.$
Thus, $ 1\iff 3.$
To prove the equivalence
$ 2\iff 3,$ it is enough to note (in the above computations) that
$$ \ffi^2(\om)\,\|\pi_{F_\om}(f)\|^2=
\ffi^2(\om) \,\|\sum_{j\in J_\om} e_{\om j}\|
= \sum_{j\in J_\om} ||^2.
$$
This finish the proof.
\med
One of the main notions in analytic theories (mathematical analysis,
functional analysis, operator ideals, operator algebras etc.) is
the notion of completeness.
Recall that a family of subspaces $ \left\{ F_\om\right\}$
is complete in a Hilbert space $H$ if
$$ \overline{span}_{\om\in\Om} \left\{ F_\om\right\}=H.
$$
The following useful fact takes a place.
\subsection*{\textbf{Proposition 2}}
Let $ \left\{ F_\om\right\}$ be a family of subspaces in
$ H,$
$ \ffi\in L^2.$
The correspondent MSF $($if MSF$)$ is complete in $ H.$
\small
{\it Proof.}\
Suppose that the family $ \left\{ F_\om\right\}$ is not complete.
Take a nonzero element $ f\in H$ such that
$ f\perp {\overline {span}}_{\om\in\Om} \left\{ F_\om\right\}.$
Then
$ \int_\Om \ffi^2 \|\pi_{F_\om}\,f\|^2=0$
(because of all $ \pi_{F_\om}\,f$ are zero).
Hence, our family is not a MSF (see (1.4)\,).
\med
A simple completeness criteria (cf. [1]):
\subsection*{\textbf{Lemma 3}}
Let $ \left\{ F_\om\right\}$ be a family of subspaces in
$ H$ and let, for each $ \om\in\Om,$
$ \left\{ e_{\om j}\right\}_{j\in J_\om}$ be an orthonormal basis in
$ F_\om.$ The following assertions are equivalent:
(1)\ $ \left\{ F_\om\right\}$ is complete in $ H;$
(2)\ $ \left\{ e_{\om j}\right\}_{\om\in \Om, j\in J_\om}$
is complete.
\med
Let us mention without a proof (which will appear in the part II
of the work) one interesting fact which is an analogue
of a result of Pete Casazza (Proposition 4 is not very trivial
assersion).
\subsection*{\textbf{Proposition 4}}
If we throw out of some MSF a subspace then we will get
either an MSF again with the same function $ \ffi$
or a noncomplete family of subspaces.
\med
Finally, a hereditary property of MSF's.
\subsection*{\textbf{Proposition 5}}
Let $ \left\{ F_\om\right\}$ be a MSF for $ H$ with a function
$ \ffi$ and with bounds $ C,D.$
If $ E$ is a subspace of a Hilbert space $ H,$ then
$ \left\{ F_\om\cap E\right\}$ is an MSF for $ E$
with the same generating function and with the same bounds.
\small
{\it Proof.}\
For $ f\in E,$ we have:
$$ \int_\Om \ffi^2 \|\pi_{F_\om}(f)\| =
\int_\Om \ffi^2 \|\pi_{F_\om\cap V}(f)\|^2.
$$
\med
In next parts (next papers) of our investigations
we will study Parseval MSFs, uniform MSFs,
Bessel measurable family of subspaces etc.,
coming later to some applications in wavelets theory.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
\bigskip
\medskip
\begin{thebibliography}{9}
\bibitem{1}
Casazza, P.~G., The art of frame theory,
Taiwanese J. Math. 4, (2000), 129-201.
\bibitem{2}
Casazza, P.~G. and Kutyniok, G., Frames of subspaces,
Contemp. Math. 345, (2004), 87-114.
\bibitem{3}
Christensen, O., An introduction to frames and Riesz bases,
Birkh{\"a}user, Boston, (2003).
\bibitem{3}
Diestel, J, and Uhl, J.~J., Vector measures,
Math. Survey 15, Amer. Math. Soc.,
Providence RI, (1977).
\end{thebibliography}
\end{document}