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%TCIDATA{Created=Wednesday, July 24, 2002 11:52:45}
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\begin{document}
Problems discussed by members of the NSF Focused Research Group on
\begin{center}
\textbf{Wavelets, Frames, and Operator Theory}
\end{center}
\noindent in the Concentration Week, July 15--20, 2002, at Texas A\&M University.
In addition to A.~Aldroubi, C.~Heil, D.~Larson, P.E.T.~Jorgensen, and
G.~\'{O}lafsson from the FRG project, many invited participants from Texas,
from around the country, from Europe, and from Singapore, took part in the
workshop.\medskip
\textbf{Manos Papadakis} (7/19/02)
Let $\left\{ V_{j}\right\} _{j}$ be a GFMRA of $L^{2}\left( \mathbb{R}%
\right) $, and assume that $M_{0}$ is the low-pass filter associated with it.
It is true that the $M_{0}\left( \xi\right) $'s define a bounded operator
$M$ for $\mathrm{a.e.}\;\xi\in\left[ 0,1\right) $, and that
$\operatorname*{ess}\,\sup\left\{ \left\Vert M\left( \xi\right) \right\Vert
:\xi\in\left[ 0,1\right) \right\} <+\infty$. Is it true that the function
$\xi\rightarrow\left\vert M_{0}\left( \xi\right) \right\vert $ is a low-pass
filter for a GFMRA? An answer may open the way to proving the connectivity of
orthogonal wavelets, because all of them are associated with GFMRA's.\medskip
\textbf{Dave Larson} (7/20/02):
Let $\psi_{H}^{{}}$ be the Haar wavelet in $L^{2}\left( \mathbb{R}\right) $.
Let $\eta\left( t\right) =\psi_{H}^{{}}\left( t-1\right) $, the
$1$-translate. Can $\psi_{H}^{{}}$ and $\eta$ be connected by a continuous
path in the $L^{2}$ metric so that all points in the path are real-valued
orthonormal wavelets? Since the winding number jumps from $\psi_{H}^{{}}$ to
$\eta$, we know from \cite[Ch.\ 2]{BrJo02} that such a homotopy path, real or
complex, \emph{cannot} be made with MRA-wavelets that have Lipschitz filters
$m_{i}\colon\mathbb{T}\rightarrow\mathbb{C}$.
\textbf{An illustration} for D.~Larson's question:%
\[%
\begin{array}
[c]{ccc}%
\text{Haar} & & \text{translate}\\
\psi_{H}^{{}} & & \psi_{H}^{{}}\left( \,\cdot\,-1\right) =\eta\left(
\,\cdot\,\right) \\%
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\end{array}
\]
Hence%
\[
\int x\left\vert \psi_{H}^{{}}\left( x\right) \right\vert ^{2}\,dx=\frac
{1}{2}\text{\quad and\quad}\int x\left\vert \psi_{H}^{{}}\left(
\,\cdot\,-1\right) \right\vert ^{2}\,dx=\frac{1}{2}+1.
\]
Connect $\psi_{H}^{{}}$ and $\eta=\psi_{H}^{{}}\left( \,\cdot\,-1\right) $
with a homotopy path in $L^{2}\left( \mathbb{R}\right) $, real-valued.
Describe the filters $\left\{ m_{t}\mid0\leq t\leq1\right\} $, $m_{0}%
\sim\psi_{H}^{{}}$, $m_{1}\sim\psi_{H}^{{}}\left( \,\cdot\,-1\right)
=\eta\left( \,\cdot\,\right) $.\medskip
\textbf{A.~Aldroubi}---as recounted by P.~Jorgensen
A thought: A law of large numbers for wavelets? Let some nice wavelet $\psi$
be given. Then establish the following \textquotedblleft limit
laws\textquotedblright:
\begin{enumerate}
\item $\operatorname*{Norm}\nolimits_{n}(\underbrace{\psi\ast\cdots\ast\psi
}_{n\text{ times}}%
)\xrightarrow[n\to\infty]{L^{2}(\mathbb{R})\text{ (other?)}}\psi
_{\text{Shannon}}^{{}}$, where \textquotedblleft$\operatorname*{Norm}%
$\textquotedblright\ means some renormalization, and
\item Let $\psi_{D_{n}}^{{}}$ denote the Daubechies wavelets given by the
Daubechies polynomials $P_{n}$ \cite{Dau92}. Show $\psi_{D_{n}}^{{}%
}\xrightarrow[n\to\infty]{L^{2}(\mathbb{R})\text{? (other?)}}\psi
_{\text{Shannon}}^{{}}$. Recall $\hat{\psi}_{\text{Shannon}}^{{}}%
=\chi_{\left( -\pi,-\frac{\pi}{2}\right) \cup\left( \frac{\pi}{2}%
,\pi\right) }^{{}}$.\medskip
\end{enumerate}
\textbf{Radu Balan} 7/19/2002
I. \emph{Issue:} Assume $A=\left( a_{ij}\right) _{i,j\in\mathbb{Z}}$ is a
bounded operator on $\ell^{2}\left( \mathbb{Z}\right) $ and:
\begin{enumerate}
\renewcommand{\theenumi}{\roman{enumi}}
\item It is invertible, say $B=\left( b_{ij}\right) _{i,j}=A^{-1}$,
\item $\exists\,r\in\ell^{1}\left( \mathbb{Z}\right) $ such that $\left\vert
a_{ij}\right\vert \leq r\left( i-j\right) $.
\end{enumerate}
\emph{Question:} Then: $\exists\,s\in\ell^{1}\left( \mathbb{Z}\right) $ such
that $\left\vert b_{ij}\right\vert \leq s\left( i-j\right) $.
\emph{Comments:} If true, it would be a noncommutative extension of Wiener's lemma.
What is known:
\begin{enumerate}
\item Wiener lemma: The above holds true for Toeplitz $A$.
\item Twisted Wiener lemma: OK for $a_{ij}=c_{i-j}e^{i\left( \varphi
_{i}-\varphi_{j}\right) }$ (see \cite[Ch.~13]{Gro01}, \cite{FeGr97}).
\item Other classes of matrices $\dots$?
\item If instead of $\ell^{1}\left( \mathbb{Z}\right) $, one uses polynomial
decay, the result is true and known as Jaffard's lemma.
\item See also symmetric (noncommutative) normed algebras of operators $\dots
$.\medskip
\end{enumerate}
II. Find more uncertainty results for wavelet orthonormal bases.
\emph{Known:} (\cite{Bat97})
\begin{gather*}
\left\{ \psi_{mn}\right\} \text{ orthonormal basis}\\
\Downarrow\\
\int_{\mathbb{R}}\left( x-x_{0}\right) ^{2}\left\vert \psi\left( x\right)
\right\vert ^{2}\,dx\cdot\int_{\mathbb{R}}\xi^{2}\left\vert \hat{\psi}\left(
\xi\right) \right\vert ^{2}\,d\xi\geq\frac{3}{2}%
\end{gather*}
with $\frac{3}{2}$ instead of $\frac{1}{2}$ (for example $x_{0}=\int
_{\mathbb{R}}x\left\vert \psi\left( x\right) \right\vert ^{2}~dx$).
(\cite{Bal98}) Similar if $\left\{ \psi_{mn}\right\} $ Bessel
sequence.\medskip
\textbf{Chris Heil} July 20, 2002
\emph{Q1.} Are the continuous wavelets path-connected in the $L^{\infty}$-norm?
(For the one-parameter family of examples, see the applet\ of Wim Sweldens
\cite{SwAp97}, and C.~Heil's papers \cite{HeSt95}, \cite{Hei94}, \cite{CoHe92}.)
\emph{Q2.} Open problem due to R. Zalik: Does $\exists\,g\in L^{2}\left(
\mathbb{R}\right) $ and countably many points $\left\{ a_{n}\right\}
_{n=1}^{\infty}$ such that%
\begin{equation}
\left\{ g\left( x-a_{n}\right) \right\} _{n=1}^{\infty} \tag{$\ast$}%
\end{equation}
is a Schauder basis for $L^{2}\left( \mathbb{R}\right) $?
\emph{Remark:} It is known that a system ($\ast$) consisting only of
translations cannot form a frame or Riesz basis for $L^{2}\left(
\mathbb{R}\right) $, but the question for Schauder basis seems to be much
more delicate.
\emph{Q3.} If $\left\{ 2^{\frac{n}{2}}\psi\left( 2^{n}x-k\right) \right\}
_{n,k\in\mathbb{Z}}$ is a wavelet frame that is not a Riesz basis, is it
finitely linearly independent (i.e., is every finite subset still linearly independent)?
\emph{Q4.} Are there \textquotedblleft nice\textquotedblright\ necessary or
sufficient conditions that ensure that a given scaling function is continuous?
\emph{Remark:} For necessary \emph{and} sufficient it is known that continuity
is determined by the \emph{joint spectral radius} of certain matrices, but
this JSR is difficult to compute in general.\medskip
\textbf{Gitta Katyniak} July 20, 2002
\texttt{gittak@upb.de}
\emph{Question 1:} Let $\Lambda\subseteq\mathbb{R}^{2}$ be discrete, and
suppose $\mathcal{D}^{-}\left( \Lambda\right) =1$, where $\mathcal{D}%
^{-}\left( \Lambda\right) $ is the lower density of $\Lambda$ When does
there exist $\psi\in L^{2}\left( \mathbb{R}\right) $ such that
\[
\left\{ x\mapsto e^{2\pi ibx}\psi\left( x-a\right) \right\} _{\left(
a,b\right) \in\Lambda}%
\]
is a frame for $L^{2}\left( \mathbb{R}\right) $? Or in other terms:
Characterize those discrete sets $\Lambda\subseteq\mathbb{R}^{2}$ with
$\mathcal{D}^{-}\left( \Lambda\right) =1$ such that there exists $\psi\in
L^{2}\left( \mathbb{R}\right) $ so that
\[
\left\{ x\mapsto e^{2\pi ibx}\psi\left( x-a\right) \right\} _{\left(
a,b\right) \in\Lambda}%
\]
is a frame for $L^{2}\left( \mathbb{R}\right) $!
\emph{Question 2:} Let $\Lambda\subset\mathbb{R}^{+}\times\mathbb{R}$ be
discrete and $\mathcal{D}_{\text{aff}}^{-}\left( \Lambda\right) >0$.
($\mathcal{D}_{\text{aff}}^{-}\left( \Lambda\right) $ is the density with
respect to the affine group.) Does there always exist $\psi\in L^{2}\left(
\mathbb{R}\right) $ such that
\[
\left\{ x\mapsto\left\vert a\right\vert ^{-\frac{1}{2}}\psi\left( \frac
{x}{a}-b\right) \right\} _{\left( a,b\right) \in\Lambda}%
\]
is a frame for $L^{2}\left( \mathbb{R}\right) $?\medskip
\textbf{Remarks of P.~Jorgensen on connectivity of wavelets:} The answer to
the question of connectivity for wavelets depends on the context: MRA
wavelets, GMRA wavelets, a connecting path in $L^{2}\left( \mathbb{R}\right)
$, a connecting path within some specified family of wavelet filters. There
may be a path in $L^{2}\left( \mathbb{R}\right) $ which connects two
\textquotedblleft nice\textquotedblright\ wavelets, but the path takes you
outside the wavelet filters which satisfy some \textquotedblleft
mild\textquotedblright\ regularity condition, such as a Lipschitz property. A
case in point is represented by $\psi_{H}^{{}}$ and its translate $\psi
_{H}^{{}}\left( \,\cdot\,-1\right) $ where%
\[
\psi_{H}^{{}}\left( x\right) =\left\{
\begin{array}
[c]{ll}%
1, & 0\leq x<\frac{1}{2},\\
-1, & \frac{1}{2}\leq x<1,\\
0, & \text{all other }x\in\mathbb{R}.
\end{array}
\right.
\]
For more details see \cite{BrJo02}, \cite{Gar98}, \cite{Gar99}, \cite{BGRW99},
\cite{WUTAM}.
For the \textquotedblleft four-tap wavelet family\textquotedblright%
\ (W.~Sweldens applet \cite{SwAp97}) it is known that the continuous wavelets
are connected \cite{HeSt95}, \cite{Hei94}, \cite{CoHe92}, and because of the
joint spectral radius characterization of continuity, that the continuous ones
form an open subset, but whether it is connected or not is completely open:
the structure of the family of $4$-tap examples is quite different from that
of the $6$-tap examples; see \cite[Ch.~2]{BrJo02}. \emph{Terminology:}
\textquotedblleft$4$-tap\textquotedblright\ refers to masking coefficients
$a_{0}$, $a_{1}$, $a_{2}$, $a_{3}$, and \textquotedblleft$6$%
-tap\textquotedblright\ refers to $a_{0}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$,
$a_{5}$.
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