text:
\section*{Introduction}

The discrete Gabor transform is a useful tool for the analysis and synthesis of nonstationary signals. It is based on the representation of the energy distribution of a signal in the time-frequency plane. Its applications range over the decomposition of musical and acoustical signals \cite{Doerf:2001}, \cite{Russetal:1998}, wireless communication \cite{HlawatschMatz:2011}, \cite{Srirametal:2013} and to the analysis of EEG signals \cite{Blancoetal:1996}, \cite{Chenetal:2010}.
For a given window function $g\in L^2(\R)$ and lattice parameters $\alpha,\beta>0$, the system of all corresponding time-frequency shifts
$$\gab =\{ M_{l\beta}\,T_{k\alpha}\,g = e^{2\pi il\beta\cdot}\,g(\cdot -k\alpha) \mid k,l\in\Z \}$$
is called a Gabor system for $L^2(\R)$. It is called a Gabor frame for
$L^2(\R)$, if  there exist constants $A,B>0$, such that
\begin{equation}\label{eq:framecond}
   A\|f\|^2 \le \sum_{k,l\in\Z} |\langle f, M_{l\beta} T_{k\alpha} g\rangle|^2
	\le B\|f\|^2,\quad \forall f\in L^2(\R) .
\end{equation}
 The constants $A,B$ are
 called lower and upper frame bounds of $\mathcal{G}(g, \alpha , \beta
 )$. If $\mathcal{G}(g, \alpha , \beta
 )$  fulfills only  the
right hand inequality,  it is called a Bessel sequence and $B$ a Bessel
bound. It is known that the frame inequality~\eqref{eq:framecond}
implies the  existence of  a dual
Gabor frame $\mathcal G(\gamma,\alpha,\beta)$  with dual window
$\gamma\in L^2(\R)$, such that every $f\in L^2(\R)$ can be represented
as
\begin{align}\label{dual:recon}
f = \sum_{k,l\in\Z}\sprod{f}{M_{l\beta}T_{k\alpha}g}\,M_{l\beta}T_{k\alpha}\gamma.
\end{align}
For a given Gabor system, the Gabor transform of a signal $f$ is defined as the analysis operator
\begin{align*}
&\mathcal C_g:L^2(\R)\rightarrow\ell^2(\Z^2),\\
&\mathcal C_gf = (\sprod{f}{M_{l\beta}T_{k\alpha}g})_{k,l\in\Z}.
\end{align*}
The  coefficient $\sprod{f}{M_{l\beta}T_{k\alpha}g}$  represents the
energy distribution of $f$ near the point $(k \alpha , l \beta )$ in
the time-frequency plane. It may also be interpreted as the amplitude
of the frequency $l\beta $ at time $k\alpha $, insofar as such an
interpretation is compatible with the uncertainty principle.

The associated synthesis operator for the reconstructions
\eqref{dual:recon} is the adjoint operator $\mathcal C_g^*$,  and the
frame operator $\mathcal S_g:L^2(\R)\rightarrow L^2(\R)$ is defined by
\begin{align*}
\mathcal S_gf = \mathcal C_g^*\,\mathcal C_gf = \sum_{k,l\in\Z}\sprod{f}{M_{l\beta}T_{k\alpha}g}\,M_{l\beta}T_{k\alpha}g.
\end{align*}

In general, there exist many dual windows suitable for the
reconstruction~\eqref{dual:recon}. The standard choice is the
canonical dual window $\gamma ^\circ = \mathcal{S}_g ^{-1} g$. For a
characterization of all dual windows
see~\cite{Chris:2003,Groech:2001}. Since the applicability and
usefulness of Gabor frames depends heavily on the knowledge and
computability of a dual window,  the numerical construction of dual
windows has motivated numerous studies. As representative
contributions we mention ~\cite{JansSon:2007,strohmer98} and the large
time-frequency analysis toolbox (LTFAT)~\cite{ltfatnote030}.

Our contribution to the analysis of dual Gabor windows is twofold. On a general level, we study  numerically
stable methods for the computation of the canonical dual  window $\gamma^\circ =
\mathcal S_g^{-1}g$. On a specific level,
we study the efficient construction and the behavior of a sequence of
dual windows for Gabor frames with  totally positive window
functions and with exponential B-splines.

We first  present two stable implementations of
a conventional iterative algorithm to approximate $\gamma^\circ$. The
algorithm was originally proposed by Schulz \cite{Schulz:1933} for
matrix inversion. It is based on the Neumann series for the inverse
frame operator and converges quadratically, see Algorithm
\ref{Schulz_iter}. We provide a detailed analysis of the numerical
error and show that our two implementations (the operator version and
the first vector version) are stable. By contrast, the
implementation proposed by Janssen (see \cite{Janssen:2002} and
\cite{Janssen:2003.3}) is often unstable  because  the numerical
error roughly doubles in each step. Therefore  the first two
implementations are much preferable. As two illustrative examples, we
use the window functions MONSTER defined in \cite{JansSon:2007} and a
Gaussian window $g$, in order to compare all three
implementations. The numerical results agree precisely with the
predicted behavior of the numerical error.
\vspace{.3cm}

We then describe recent results on special Gabor
systems whose window function is a totally positive (TP) function of
finite type or an exponential B-spline (EB-spline). TP functions  are
remarkable because so far they are the only window functions for which
a complete characterization of  all lattice parameters such that
$\mathcal{G}(g,\alpha,\beta )$ is a frame is known. More precisely,
 the Gabor system $\gab$, with  a TP
function $g$ of finite type $N\ge 2$ is  a  frame if and only if
$\alpha\beta<1$~\cite{GroeStoe:2013}. Subsequently, similar arguments
in~\cite{KloStoe:2014} showed   that the Gabor
system $\mathcal G(B_\Lambda,\alpha,\beta)$ of an EB-spline
constitutes a frame for $\alpha = 1$, $\beta<1$,  and some other lattice
parameters, too. The proofs also provide a constructive method for the
computation of infinitely many dual windows $\gamma_L$ with compact
support, which we summarize in Algorithm~\ref{algo:gamma} for TP
functions of finite type and Algorithm~\ref{algo2:gamma} for
EB-splines.

This construction offers several new and useful aspects that are special
for TP windows and not shared by general window functions.

(i) Algorithms~\ref{algo:gamma} and ~\ref{algo2:gamma} provide a
family of dual windows $\gamma _L$ both in finite and infinite
dimensional models, namely for
continuous signals in $L^2(\R )$, for discrete signals in $\ell ^2(\Z
)$, and for periodic discrete signals in $\C ^N$. Currently available
toolboxes, such as LTFAT~\cite{ltfatnote030},  work only for finite-dimensional
signals.

(ii) The dual windows $\gamma _L$ possess compact support of size
$\mathcal{O}(L)$, whereas the canonical dual $\gamma ^\circ $ is known
to have infinite support.

(iii) The dual windows $\gamma _L$ are exact and
satisfy~\eqref{dual:recon}. This is  in contrast to the standard
iterative methods for the approximation of the canonical dual (see
e.g. Algorithm~\ref{Schulz_iter}), which generate only approximations
of a dual  window.

As our main mathematical result
we prove that the dual windows
$\gamma_L$ are good approximations of the canonical dual window
$\gamma^\circ = \mathcal S_g^{-1}g$ and we show that they converge
exponentially fast to the canonical dual window, i.e.,
$\|\gamma_L-\gamma^\circ\|_2=\mathcal{O}(e^{-\rho L})$.
Therefore, by specifying the
parameter $L$, Algorithms~\ref{algo:gamma} and \ref{algo2:gamma}
provide a dual window $\gamma_L$ with compact support and which
approximates the canonical dual at a desired rate.

 The proof uses some ideas of the non-symmetric finite
section method, but also requires a new technique related to the
formulation of the Moore-Penrose pseudo-inverse of infinite matrices
in terms of orthogonal projections.
\vspace{.3cm}

As our main numerical contribution, we study and implement
the case of discrete Gabor frames. We present some fast and stable
algorithms to evaluate and discretize TP functions and EB-splines and
their  dual windows
computed by the Algorithms~\ref{algo:gamma} and
\ref{algo2:gamma}. These algorithms are proposed as extensions to the
Large Time Frequency Analysis Toolbox described in
\cite{ltfatnote030}.

The paper is organized as follows: In section~\ref{sec:hotelling} we
study the numerical stability of a fast iterative  algorithm for the
approximation of the canonical dual window. In section~\ref{sec:known}
we summarize the algorithms for the construction of dual windows of TP
functions and EB splines. In section~\ref{sec:approx} we formulate and
discuss the main theorems about the convergence of the compactly
supported dual windows $\gamma _L$ to the canonical dual window $\gamma
^\circ $. Section~\ref{sec:discrete} explains some details about the
implementation of Gabor frames with TP functions and EB splines.
The appendix contains the technical details of the
proofs of the main results.

\section{Some iterative algorithms for approximating the canonical dual}\label{sec:hotelling}

In this section we describe two iterative algorithms for approximating
the canonical dual of an arbitrary frame $\mathcal F=\{f_j\}_{j\in I}$
for a Hilbert space $\mathcal H$. The central part of such algorithms
is the approximation of the inverse of the corresponding frame
operator
$$S_{\mathcal F}:\mathcal H\to \mathcal H,\qquad
S_{\mathcal F} h=\sum_{j\in I} \langle h,f_j\rangle f_j.$$
We  discuss the convergence and the
numerical stability of various implementations. Finally we present
some numerical tests. The following approximation schemes are proposed
in the literature.
\begin{algorithm}[Frame algorithm]
Choose $0<\lambda<2/B$, with $B$ the upper frame bound of $\mathcal F$. Then $q:=\norm{I-\lambda\,S_{\mathcal F}}<1$ and
$$S_{\mathcal F}^{-1} = \lambda\,\sum_{n=0}^\infty (I-\lambda\,S_{\mathcal F})^n.$$
The partial sums of this Neumann series can be computed iteratively  by
\begin{align}
&K_0 = \lambda\,I, \notag \\
&K_{k+1} = \lambda\,I + (I-\lambda\,S_{\mathcal F})\,K_k,\ \
k\in\N_0. \label{eq:cmr}
\end{align}
The convergence rate is  of order $\norm{S_{\mathcal F}^{-1}-K_{k}} =
\mathcal O(q^k)$.
The $k$'th approximation of the canonical dual frame $\mathcal F^\circ
= \{S_{\mathcal F}^{-1}\,f_j\}_{j\in I}$ is given by
$\{K_k\,f_j\}_{j\in I}$.
\end{algorithm}
This algorithm is very robust, but slow if $q$ is close to one.
This algorithm can be accelerated~\cite{gro93ieee} with
conjugate gradient techniques and with a convergence rate
$\big(\frac{\sqrt{B}- \sqrt{A}}{\sqrt{B}+ \sqrt{A}}\big)^k$ after $k$
iterations regardless of whether the frame bounds $A,B$ are known or
not.

An even faster method goes back to Schulz~\cite{Schulz:1933} and
Hotelling~\cite{Hotelling:1942}.

\begin{algorithm}[Schulz iteration]\label{Schulz_iter}
 Choose  $0<\lambda < 2/B$.  The version of Schulz iteration  with
 "initial scaling"~\cite[Algorithm~IV]{Janssen:2003.3} is
\begin{align}\notag
&J_0 = \lambda\,I,\\\label{Schulz_op}
&J_{k+1} = 2\,J_k - J_k\,S_{\mathcal F}\,J_k,\ \ k\in\N_0.
\end{align}
This iteration implies  the identity $J_k = K_{2^k-1}$ and is
therefore connected  to the frame algorithm. The Schulz algorithm  converges quadratically, i.e.
\begin{equation}
  \label{eq:c13}
\norm{S_{\mathcal F}^{-1}-J_{k+1}} \le \norm{S_{\mathcal
    F}}\,\norm{S_{\mathcal F}^{-1}-J_k}^2 = \mathcal{O}(q^{2^{k+1}}) .
\end{equation}
\end{algorithm}
This algorithm was first described by Schulz \cite{Schulz:1933}  who used this method   for matrix inversion.
\begin{IEEEproof}
The claims in Algorithm~\ref{Schulz_iter} are proved by induction. Since Schulz's algorithm is not as known as other iterative
algorithms, we sketch the main steps.
We first show that
\begin{equation}
  \label{eq:c19}
  I - S_\mathcal{F}J_k  = I - J_k S_\mathcal{F} = (I - \lambda S_\mathcal{F}
  )^{2^k} \, .
\end{equation}
Assuming that \eqref{eq:c19} is correct for $k\in \mathbb{N}_0$, we
obtain
\begin{align*}
  I - S_\mathcal{F}J_{k+1} &= I - S_\mathcal{F} (2 J_k - J_k S_\mathcal{F} J_k) \\
&= (I - S_\mathcal{F}J_k)^2 = \Big( (I - \lambda S_\mathcal{F} )^{2^k} \Big)^2
\, ,
\end{align*}
as claimed.
Using~\eqref{eq:c19}, we show again by induction that  $J_k = K_{2^k-1} =\lambda \sum
_{j=0}^{2^k-1} (I - \lambda S_\mathcal{F} )^j $:
\begin{align*}
\lambda & \sum_{j=0}^{2^{k+1}-1}  (I-\lambda S_\mathcal{F} )^j  \\
                    &           = \lambda
                               \sum_{j=0}^{2^{k}-1}(I-\lambda S_\mathcal{F}
                               )^j + (I-\lambda S_\mathcal{F} )^{2^k} \, \lambda
                               \sum_{j=0}^{2^{k}-1}(I-\lambda S_\mathcal{F} )^{j}\\
                               &= J_k +  (I - J_k S_\mathcal{F} ) J_k =
                               J_{k+1} \, .
\end{align*}
The quadratic convergence rate now  follows  from the convergence
properties of the Neumann series.
\end{IEEEproof}

We discuss the implementation of Algorithm~\ref{Schulz_iter} in the
case of a Gabor frame $\mathcal G(g,\alpha,\beta)$. Recall that the
canonical dual frame is determined by the dual window
$\gamma^\circ=\mathcal S_g^{-1}g$. We compare three different
implementations of the Schulz iteration  and provide some heuristics
for their  numerical
stability.
\medskip

\noindent (i) \emph{Operator form}:  The numerical computation of the
  Schulz iteration as stated in Algorithm~\ref{Schulz_iter} provides
  operators $\hat{J}_k = J_k + E_k$, where $E_k$ denotes the
  accumulated forward error. Let  $Y_{k+1}$ denote the new roundoff
  error in the $k+1$'st iteration, then the operator  after  $k+1$
  iterations of  (\ref{Schulz_op}) is
\begin{align*}
\hat{J}_{k+1} &= 2\,\hat{J}_k - \hat{J}_k\,\mathcal S_g\,\hat{J}_k + Y_{k+1}\\
              &= J_{k+1} + E_k\,(I-\mathcal S_g\,J_k) +
              (I-J_k\,\mathcal S_g)\,E_k

              + Y_{k+1} +
              \mathcal O(\norm{E_k}^2).
\end{align*}
 Since $I-\mathcal S_g\,J_k = I-J_k\,\mathcal S_g =
(I-\lambda\,\mathcal S_g)^{2^k}$,  we have
$$\norm{E_{k+1}} = \norm{\hat{J}_{k+1} - J_{k+1}}  \le 2\,q^{2^k}\,\norm{E_k} + \norm{Y_{k+1}} + \mathcal O(\norm{E_k}^2).$$
This estimate shows that the error accumulated in the first $k$
iterations is damped and only a new round-off error is added.  Hence this iteration is numerically stable.
\medskip

\noindent (ii) \emph{ Vector form}:  We compute approximations of the
  dual window $\gamma^\circ$ directly by setting $\gamma_k := J_k\,g$
  with the following algorithm:
\begin{align}\notag
&\gamma_0    = \lambda\, g \\\label{Hotelling_el}
&\gamma_{k+1}= 2\, \gamma_k - \mathcal C^*_{\gamma_k}\,\mathcal
C_{g}\,\gamma_k,\ \ k\in\N_0 \, .
\end{align}
\begin{IEEEproof}
We use  the fact that $S_\mathcal{F}$ and thus all
$J_k$ commute with the time-frequency shifts $M_{l\beta } T_{j\alpha
}$. This commutation rule implies  the  identity
\begin{align*}
\mathcal C_{\gamma_k}^*\,\boldsymbol{c} = \sum_{j,l\in
  \Z}c_{j,l}\,J_k\,M_{l\beta}\,T_{j\alpha}\,g = J_k\,\mathcal
C_g^*\,\boldsymbol{c}
\end{align*}
for all $\boldsymbol{c}\in\ell^2(\Z^2)$.  Consequently
 \begin{align*}
   \gamma _{k+1} &= J_{k+1} g = 2 J_k g -  J_k \mathcal{S}_g J_kg \\
   &= 2\gamma _k -  J_k \mathcal{C}^*_g \mathcal{C}_g \gamma_k = 2\gamma _k
   -  \mathcal{C}^*_{\gamma _k} C_g \gamma _k \, .
 \end{align*}
\end{IEEEproof}
 The numerical computation yields  $\hat{\gamma}_k = \gamma_k + e_k$,
 where $e_k$ denotes the accumulated forward error. Let  $y_{k+1}$
 denote the new roundoff error, then in  the $k+1$'th
 step of the iteration (\ref{Hotelling_el})  we have
\begin{align*}
\hat{\gamma}_{k+1} &= 2\,\hat{\gamma}_k - \mathcal C^*_{\hat{\gamma}_k}\,\mathcal C_{g}\,\hat{\gamma}_k + y_{k+1}\\
              &= \gamma_{k+1} + (e_k-\mathcal C_{\gamma_k}^*\,\mathcal
              C_g \,e_k) + (e_k - \mathcal  C_{e_k}^*\,\mathcal
              C_g\,\gamma_k)

              + y_{k+1} + \mathcal
              O(\norm{e_k}_2^2) \, .
\end{align*}
 The aforementioned estimates give
$$\norm{(I-\mathcal C_{\gamma_k}^*\,\mathcal C_g )\,e_k}_2 = \norm{(I-J_k\,\mathcal S_g)\,e_k}_2\le q^{2^k}\,\norm{e_k}_2.$$
Moreover the Janssen representation (see \cite[p.131]{Groech:2001}) gives
$$\mathcal C_{e_k}^*\,\mathcal C_g\,\gamma_k = \frac{1}{\alpha\beta}\sum_{j,l\in\Z}\sprod{e_k}{M_{j/\alpha}\,T_{l/\beta}\,g}M_{j/\alpha}\,T_{l/\beta}\,\gamma_k.$$
The last expression, when $\gamma_k$ is replaced by $\gamma^\circ$, is
the orthogonal projection $\Pi_{V_g} e_k$ of $e_k$ onto $V_g :=
\overline{\mathrm{span}}\bigl(\mathcal G(g,1/\beta,1/\alpha)\bigr)$
and therefore
$$\norm{e_k - \mathcal  C_{e_k}^*\,\mathcal
  C_g\,\gamma^\circ}_2\le\norm{e_k}_2$$
and
\begin{align*}
\norm{e_k &- \mathcal  C_{e_k}^*\,\mathcal
  C_g\,\gamma _k}_2 \le \norm{e_k - \mathcal  C_{e_k}^*\,\mathcal
  C_g\,\gamma^\circ}_2

  +\norm{C_{e_k}^*\,\mathcal
  C_g\,(\gamma^\circ - \gamma _k)}_2 \leq \norm{e_k}_2 +
\mathcal{O}(q^{2^k}) \, .
\end{align*}
Since $\norm{\gamma^\circ - \gamma_k}_2=\mathcal O(q^{2^k})$ by
\eqref{eq:c13},  the new numerical error is
$$\norm{e_{k+1}}_2 \leq (1+q^{2^k}) \norm{e_k}_2 +
  \norm{y_{k+1}}_2 + \mathcal{O}(\norm{e_k}^2_2 + q^{2^k})  \, .
$$
In contrast to the operator version, $\norm{e_k}_2$ enters linearly
with coefficient $\approx 1$. Thus
the numerical stability is plausible and  is also
confirmed by our numerical results in Example~\ref{numerical_results}.
\medskip

\noindent (iii) \emph{Janssen's alternative vector version}:  Janssen \cite[Algorithm~IV]{Janssen:2003.3}
  proposes  the
  approximations
\begin{align}\notag
&\gamma_0    = \lambda\, g, \\\label{Janssen_el}
&\gamma_{k+1}= 2\, \gamma_k - \mathcal C^*_{\gamma_k}\,\mathcal C_{\gamma_k}\,g,\ \ k\in\N_0.
\end{align}
However,  in their numerical tests of \eqref{Janssen_el}   Janssen and S{\o}ndergaard \cite{JansSon:2007}   observed some numerical instability. With notations as in (ii), the numerical error in step $k+1$ is
\begin{align*}
  e_{k+1}&=\hat{\gamma}_{k+1}-\gamma_{k+1} \\
&=2\,e_k - \mathcal
  C_{\gamma_k}^*\,\mathcal C_{e_k}\,g - \mathcal C_{e_k}^*\,\mathcal
  C_{\gamma_k}\,g  + y_{k+1} + \mathcal{O}(\norm{e_k}_2^2).
\end{align*}
We show that the error may grow by at least a factor of $2$ in each step. Note that $V_g$ in (ii) is a proper subset of $L^2(\R)$, if $\alpha\beta<1$. Hence $\norm{(I-\Pi_{V_g})\,e_1}_2\approx \varepsilon$, where $\varepsilon$ denotes the machine accuracy. The Janssen representation implies, that $\mathcal C_{\gamma_k}^*\,\mathcal C_{e_k}\,g$ and $\mathcal C_{e_k}^*\,\mathcal C_{\gamma_k}\,g$ are in $V_g$. Therefore,
\begin{align*}
(I-\Pi_{V_g})\,e_{k+1} = 2\,(I-\Pi_{V_g})\,e_k&+(I-\Pi_{V_g})\,y_{k+1}

   +\mathcal O(\norm{e_k}_2^2)
\end{align*}
This shows, that the error components in $V_g^\perp$ can double in
each step.  In Example~\ref{numerical_results} we demonstrate that
this numerical instability  may indeed occur.

\begin{center}
\begin{figure}
\hspace*{4.5cm
\includegraphics[width=8.8cm]{monster.eps}
\caption{MONSTER function and its canonical dual for $\alpha=20$ and $\beta=1/50$, computed by a routine of LTFAT \cite{ltfatnote030}.}\label{plot:monster}
\end{figure}
\end{center}

\begin{example}\label{numerical_results}
As it was described in (iii) the implementation (\ref{Janssen_el}) by
Janssen can have some stability problems and should be applied
carefully. We use the function MONSTER (see Figure~\ref{plot:monster})
in \cite{JansSon:2007} and $\alpha=20$, $\beta=1/50$ for our numerical
tests of all three implementations of the Schulz iteration. As we can
see in Figure~\ref{plot:error}, the operator version is stable, the
vector version of (ii) is also useful, while the error of the
implementation in (iii) explodes. The same conclusions hold for the window function  $g(x) =
e^{-\pi x^2/600}$ with  $\alpha= 20, \beta= 1/50$,  as is shown in Figure~\ref{plot:error2}.
\end{example}

\begin{center}
\begin{figure}
\hspace*{4.5cm
\includegraphics[width=8.8cm]{iter_error.eps}

\caption{Error $\norm{\mathcal{C}_{\gamma^\circ}-\mathcal{C}_{\hat\gamma_k}}$ of the three implementations of Algorithm \ref{Schulz_iter} for $20$ iteration steps to approximate the canonical dual window of MONSTER for $\alpha=20$ and $\beta=1/50$.}\label{plot:error}
\end{figure}
\end{center}

\begin{center}
\begin{figure}[h]
\hspace*{4.5cm
\includegraphics[width=8.8cm]{iter_error2.eps}

\caption{$\norm{\mathcal{C}_{\gamma^\circ}-\mathcal{C}_{\hat\gamma_k}}$ of the three implementations of Algorithm \ref{Schulz_iter} for $20$ iteration steps to approximate the canonical dual window of the Gaussian $g(x) = e^{-\pi x^2/600}$ for $\alpha=20$ and $\beta=1/50$.}\label{plot:error2}
\end{figure}
\end{center}

\newpage
\section{Gabor frames of totally positive functions and exponential B-splines}
\label{sec:known}

 We now consider totally positive functions of finite type and
 exponential B-splines as window functions. The TP functions attracted
 much interest recently, as they provide new examples of window
 functions for which the necessary density condition $\alpha\beta<1$
 of the lattice parameters is also sufficient
 \cite{GroeStoe:2013}. For detailed information on total positivity of
 functions and matrices see \cite{Karl:1968}, for a detailed
 introduction to exponential B-splines see \cite{Schum:1981} and
 \cite{deVLor:1993}.

\begin{definition}{\bf \cite{Schoenberg:1947}, \cite{Schoenberg:1951}}
An integrable function $g:\R\rightarrow\R$ is called totally positive
(TP), if
its Fourier transform factors as
\begin{align}\notag
\hat{g}(\omega) &=  \int_{-\infty}^{\infty} g(t) e^{-2\pi it\omega}\,dt \\ \label{TPall}
&= C e^{-\eta\omega^2} e^{-2\pi i\delta\omega}
\prod_{\nu=1}^{\infty}\frac{e^{2\pi i\delta_{\nu}\omega}}{1+2\pi i\delta_{\nu}\omega},
\end{align}
where $C,\eta,\delta,\delta_{\nu}$ are real parameters with
$$C>0,\ \ \eta\geq0, \
0<\eta+\sum_{\nu=1}^{\infty}\delta_{\nu}^2<\infty\, .$$
\end{definition}

Note that in Schoenberg's terminology there also exist TP functions
that  are not integrable. Therefore the given definition  does not
include the most general case of TP functions. Here we only consider TP
functions $g_\Delta$ of finite type $N\in\N$, with $\eta =0$, $\Delta=(\delta_1,\ldots,\delta_N)$ and
\begin{equation}\label{eq:TPFFT}
    \widehat{g_\Delta}(\omega) = \prod_{\nu = 1}^N(1+2 \pi i\delta_\nu\omega)^{-1}.
\end{equation}
For $N=1$ and $\Delta=(\delta)$ we have the one-sided exponential
$$
  g_\delta(x)=\frac{1}{|\delta|} \,
  e^{-\delta^{-1} x}\, \chi_{(0,\infty)}(\delta x)\quad\hbox{for}\quad x\in\R\setminus\{0\},
$$
with support $[0,\infty)$ for positive $\delta$ and $(-\infty,0]$ for negative $\delta$, and for $N>1$ and $\Delta=(\delta_1,\ldots,\delta_N)$, $g_\Delta$ is the $N$-fold convolution
$$
   g_\Delta=
   g_{\delta_1}\ast\cdots \ast
   g_{\delta_N}.
$$
Especially, if $\delta_\nu=\delta\ne 0$ for all $1\le \nu\le N$, then
$$
  g_\Delta(x)= \frac{|x|^{N-1}}{|\delta|^N(N-1)!}\,
  e^{-\delta^{-1} x}\, \chi_{(0,\infty)}(\delta x).
$$
The functions $g_\Delta$  are nonnegative, have infinite support,  and exponential
decay, precisely, if $\tau = (\max\{\,2\pi\,\abs{\delta_\nu} \mid \nu
= 1,\ldots,N\})^{-1}$ and $\epsilon >0$, then there is a constant $c =
c(\epsilon )$ such that
\begin{align}\label{TP:decay}
   0\le g_\Delta (x)\le c\,e^{-(\tau-\varepsilon)\abs{x}}.
\end{align}
Moreover, if $\delta_1\ne \delta_N$ and
we specify $g_{\delta}(0)=1/|2\delta|$,
the recurrence relation
$$g_{\delta_1,\ldots,\delta_N}=\frac{\delta_1^{-1} g_{\delta_2,\ldots,\delta_N}-
\delta_N^{-1} g_{\delta_1,\ldots,\delta_{N-1}}}{\delta_1^{-1}-\delta_N^{-1}}
$$
holds.
For later use, we denote by $m$ (resp. $n$) the number of positive (resp. negative) parameters $\delta_\nu$.

The main result in \cite{GroeStoe:2013} shows that the Gabor system $\gab$ of a TP function of finite type $m+n\ge 2$ constitutes a Gabor frame if and only if $\alpha\beta<1$. The proof provides an algorithm for the computation of a dual window $\gamma$ with compact support. This method was adapted in \cite{BanGroeStoe:2013}, \cite{Klo:2012} in order to supply infinitely many dual windows $\gamma_L$. The computation of $\gamma_L(x+j\alpha)$, with $j\in\Z$, is performed by computing a single row of a left-inverse of the biinfinite pre-Gramian matrix
\begin{align}\label{pregram}
P_g(x) := \left( \overline{g}\left(x+\alpha j -\tfrac{k}{\beta}\right) \right)_{j,k\in\Z}.
\end{align}
The structure of the left-inverse of $P_g(x)$ heavily depends on the property that $g$ is totally positive. We include the algorithm for the reader's convenience.

\begin{algorithm} {\bf \cite{BanGroeStoe:2013}}
\label{algo:gamma}
Input parameters are the parameter vector $\Delta = (\delta _1,
\delta _2, \dots , \delta _N)$ of the
window $g$, the lattice parameters $\alpha ,\beta > 0$ with
$\alpha\beta<1$,
a parameter $L\in\bN_0$ controlling the support size of the dual window $\gamma_L$,
and a point $x\in[0,\alpha)$.

Output parameters are integers $i_1(L),i_2(L)$ and
the vector of values $\gamma_L(x+\alpha j)$,
$i_1(L)\le j\le i_2(L)$, in the support of $\gamma_L$, such that $\gamma_L:\R\rightarrow\R$ defines a dual window of $g$.

  \begin{enumerate}
  \item Set $r:=\left\lfloor \frac{1}{1-\alpha \beta} \right\rfloor$, $ k_1 = -(r+1)m$ and
      $k_2 = (r+1)n$.
  \item Set $ k_1(L) := k_1-L$ and $k_2(L) := k_2+L$,
    \begin{align*}
    i_1(L) &:= \left\lfloor \frac{k_1-L + m -
        1}{\alpha\beta}-\frac{x}{\alpha}\right\rfloor+1,\\
    i_2(L) &:=
    \left\lceil \frac{k_2+L - n +
        1}{\alpha\beta}-\frac{x}{\alpha}\right\rceil-1.
    \end{align*}
   \item Set  $ P_L(x) := (p_{j,k})_{i_1(L)\le j \le i_2(L),~ k_1(L)\le k \le k_2(L)}$, where
    \[  p_{j,k}= g \left(x+\alpha j- \frac{k}{\beta}\right).
    \]
  \item Compute the
	pseudoinverse $$P_L(x)^\dagger=(q_{k,j})_{k_1(L)\le k \le k_2(L),~ i_1(L)\le j \le i_2(L)}$$
    of $P_L(x)$.
  \item Take the row with index \(k=0\) of $P_L(x)^\dagger$. Its
    coefficients define the values of the dual window $\gamma_L$ at the
    points $\{x+\alpha j \,|\,i_1(L) \le j \le i_2(L)\}$, i.e.
    \begin{equation}\label{eq:set-gamma}
    \gamma_L(x+\alpha j):=\begin{cases} \beta\, q_{0,j}&\text{, if } i_1(L)\le j \le i_2(L),\\
		0&\text{, if } j<i_1(L)~\text{or } j>i_2(L).
		\end{cases}
    \end{equation}
  \end{enumerate}
\end{algorithm}
In particular, the support of the dual window $\gamma _L$ is contained
in the interval $[\alpha i_1(L), \alpha (i_2(L)+1)]$ of length of the
order $ \beta^{-1} (k_2 - k_1 +2L )$. Thus the parameter $L$
labels the size of the support.

A  related class of window functions  is the class of exponential B-splines. These functions are positive and have compact support, a property which is desirable in some applications.

\begin{definition}
For $\Lambda=(\lambda_1,\ldots,\lambda_N)\in\R^N$ the exponential B-spline (EB-spline) $B_\Lambda$ with knots $0,1,\ldots,N$ is given by its Fourier transform
\begin{align}\label{eq:EBspline}
\widehat{B_\Lambda}(\omega) = \prod_{\nu=1}^N \frac{e^{\lambda_\nu-2\pi i\omega}-1}{\lambda_\nu-2\pi i\omega}.
\end{align}
\end{definition}

 In \cite{KloStoe:2014}, it was shown that the Gabor system $\mathcal
 G(B_\Lambda,\alpha,\beta)$ of every EB-spline constitutes a frame for
 $\alpha = 1$, $\beta<1$ (and also  some other lattice
 parameters). Similar to the case of TP functions, the following
 algorithm provides dual windows $\gamma_L$ of the Gabor frame
 $\mathcal{G}(B_\Lambda,\alpha,\beta)$.

\begin{algorithm}~{\bf \cite{KloStoe:2014}}
\label{algo2:gamma}
Input parameters are the parameter vector $\Lambda\in\R^N$ of the
window $B_\Lambda$, the lattice parameters $\alpha ,\beta > 0$,
a parameter $L\in\bN_0$ controlling the support size of the dual window $\gamma_L$,
and a point $x\in\left[\frac{N-\alpha}{2},\frac{N+\alpha}{2}\right)$.

Output parameters are integers $i_1(L),i_2(L)$ and
the vector of values $\gamma_L(x+\alpha j)$,
$i_1(L)\le j\le i_2(L)$, in the support of $\gamma_L$, such that $\gamma_L:\R\rightarrow\R$ defines a dual window of $B_\Lambda$.

  \begin{enumerate}
  \item Set $ k_2(L) = \left\lfloor \frac{N\beta+\alpha\beta}{2(1-\alpha\beta)} \right\rfloor +1 +L$ and $k_1(L) = -k_2(L)$.
  \item Set $i_1(L)\le k_1(L)$ and $i_2(L)\ge k_2(L)$, such that
	\begin{align*}
	B_\Lambda\left(x+(i_1(L)-1)\alpha-\tfrac{k_1(L)-1}{\beta}\right)\neq 0,\\
	B_\Lambda\left(x+i_1(L)\alpha-\tfrac{k_1(L)-1}{\beta}\right) = 0,\\
	B_\Lambda\left(x+(i_2(L)+1)\alpha-\tfrac{k_2(L)+1}{\beta}\right)\neq 0,\\
	B_\Lambda\left(x+i_2(L)\alpha-\tfrac{k_2(L)+1}{\beta}\right) = 0.
	\end{align*}
	\item Set  $ P_L(x) := (p_{j,k})_{i_1(L)\le j \le i_2(L),~ k_1(L)\le k \le k_2(L)}$, where
    \[  p_{j,k}= B_\Lambda \left(x+\alpha j- \frac{k}{\beta}\right).
    \]
  \item Compute the
	pseudoinverse $$P_L(x)^\dagger=(q_{k,j})_{k_1(L)\le k \le k_2(L),~ i_1(L)\le j \le i_2(L)}$$
    of $P_L(x)$.
  \item Take the row with index \(k=0\) of $P_L(x)^\dagger$. Its
    coefficients define the values of the dual window $\gamma_L$ at the
    points $\{x+\alpha j \,|\,i_1(L) \le j \le i_2(L)\}$, i.e.
    \begin{equation}\label{eq:set-gamma2}
    \gamma_L(x+\alpha j):=\begin{cases} \beta\, q_{0,j}&\text{, if } i_1(L)\le j \le i_2(L),\\
		0&\text{, if } j<i_1(L)~\text{or } j>i_2(L).
		\end{cases}
    \end{equation}
  \end{enumerate}
\end{algorithm}

We would like to emphasize the following point: (i)  These algorithms determine the precise values of a
dual window $\gamma $ with compact support in $L^2(\bR )$ and not just
a discrete approximation in a finite-dimensional vector space, as is
done in most existing  algorithms
in~\cite{ltfatnote030,strohmer98}.

(ii) Although the problem of finding a dual window is by nature
infinite-dimensional in $L^2(\R )$, the computation of $\gamma _L$  on
a grid $\tfrac{\alpha}{M} \Z $ requires only the pseudo-inversion of
$M$ finite-dimensional matrices.

\vspace{.5cm}

\section{Approximation of the canonical dual of TP functions and EB-splines}\label{sec:approx}

We fix the parameters $\alpha,\beta>0$ of the Gabor frame and let $g$
be either a TP function of finite type with parameter set $\Delta$ or
an EB-spline with parameter set $\Lambda$. The pre-Gramian matrix
\eqref{pregram} plays a central role in the characterization of the
frame bounds $A,B$ of the Gabor system $\gab$, and in finding dual
windows $\gamma$.

In this section we address the question how the sequence of dual
windows $\gamma _L$ of Algorithms~\ref{algo:gamma} and
~\ref{algo2:gamma} are related to the canonical dual window $\gamma
^\circ$.
Our main goal  is to prove that the compactly supported dual windows
$\gamma_L$ computed by the  Algorithms~\ref{algo:gamma} and
\ref{algo2:gamma} from  \cite{BanGroeStoe:2013} and
\cite{KloStoe:2014}  approximate the canonical dual window $\gamma^\circ$ at a rate
\[
   \zweinorm{\gamma_L- \gamma^\circ} \le \tilde{c}\,e^{-\rho L},
\]
where $L$ determines the support of $\gamma_L$ and $\rho>0$.

As a first step we show that the norm of $\gamma _L$ remains bounded
as $L $ tends to $\infty $.

\begin{theorem}
  \label{thm:bounds-PL}
Let $g$ be a TP function of finite type as defined by
(\ref{eq:TPFFT}). Then there exist constants $A,B>0$ independent of
$L\in \N _0$ such that
\begin{align}\label{PLbounds}
   A \|\boldsymbol{c}\|_2 \le \|  P_L(x) \boldsymbol{c}\|_2 \le    B \|\boldsymbol{c}\|_2
\end{align}
for all $\boldsymbol{c}\in \bC^{k_2(L)-k_1(L)+1}$ and $x\in [0,\alpha)$, where $P_L(x)$ is the finite section of the corresponding biinfinite pre-Gramian matrix $P_g(x)$ as described in Algorithm~\ref{algo:gamma}. Consequently, $\zweinorm{\gamma_L}\leq \sqrt{\alpha}\,\beta\,A^{-1}$, for all $L\in\N$.
\end{theorem}

The precise proof will be given in the appendix. The proof idea goes
as follows:
The upper bound $B$ is easily  obtained from Schur's test based on
the exponential decay of $g$ in (\ref{TP:decay}). For the lower bound
$A$, we choose $2L+1$ submatrices $R_\ell$ of $P_L(x)$, with
$k_2-k_1+1$ columns each, and build a left-inverse $Q_L(x)$ of
$P_L(x)$ by the selection of specific rows of $R_\ell^\dagger$. The
result in \cite[Theorem~9]{GroeStoe:2013} shows that all matrices
$R_\ell^\dagger$ are bounded uniformly in $x$. Their upper bound $C$
will provide the upper bound
$A^{-1}=C\,\tfrac{k_2-k_1+1}{\alpha\beta}$ for all matrices $Q_L(x)$
uniformly in $x$ and $L$. Hence $A$ is a suitable constant for the
lower bound of $P_L(x)$.

To prove the rate of approximation of the dual windows $\gamma_L$, we
recall   that the canonical dual window $\gamma^\circ = \mathcal
S_g^{-1}g$ can be expressed by the Moore-Penrose pseudoinverse of
$P_g(x)$, namely
\begin{equation}
  \label{eq:c15}
\gamma^\circ(x+\alpha j) = \beta\,(P_g(x)^\dagger)_{0,j}
\end{equation}
for all $j\in\Z$ and all $x\in[0,\alpha)$. This is a direct consequence of the
Wexler-Raz criterion for the dual windows of $g$ \cite[Theorem~7.3.1]{Groech:2001}, and the minimal $L^2$-norm of
the canonical dual among all duals of $g$ \cite{Janssen1995}.
We will therefore show that the zeroth row of $P_L(x)^\dagger$ approximates the zeroth
row of $P_g(x)^\dagger$ at an exponential rate. For this we need  a
new result on non-symmetric finite sections of biinfinite
matrices. The following theorem is  not covered by the results in
\cite{GroeRzeStroh:2010} and may be of independent interest.  To fix the notation, for $\boldsymbol{n} =
(n_1,n_2) \in\N^2$ and $b\in\ell^2(\Z)$, we let $\mathcal
P_{\boldsymbol{n}}$ with
$$\mathcal P_{\boldsymbol{n}}\,b = (\ldots,0,b_{-n_1},b_{-n_1+1},\ldots,b_{n_2-1},b_{n_2},0,\ldots)^T$$
be the orthogonal projection onto the $n_1+n_2+1$-dimensional subspace
$\mathcal P_{\boldsymbol{n}}\,\ell^2(\Z) \cong \C^{n_1+n_2+1}$. For a
biinfinite matrix $U=(u_{j,k})_{j,k\in\Z}$ and $\br,\bn\in\N^2$, $\mathcal
P_\br\,U\,\mathcal P_\bn$ is a non-symmetric finite section of
$U$. We  will write $(\mathcal P_\bn\,U\,\mathcal
P_\bn)^{-1}$ for the  inverse of the symmetric finite section
 $\mathcal P_\bn\,U\,\mathcal P_\bn$ on the finite-dimensional
 subspace  $\mathcal
P_\bn\,\ell^2(\Z)$ (with the understanding that it  cannot  be invertible
on $\ell^2(\Z)$).
\begin{theorem}\label{thm:fs}
Let $\big(\chi (k)\big)_{k\in\Z}$ be a strictly increasing sequence of integers and
$U=(u_{j,k})_{j,k\in\Z}$ be a biinfinite matrix such that  (a) $U^*U$ is
invertible,  and (b)  there exist constants $c,a >0$ such that
\begin{equation}\label{eq:decaying}
\abs{u_{j,k}} \leq c\,e^{-a\abs{j-\chi (k)}}\quad
\text{ for all }j,k\in\Z \, .
\end{equation}
Let $I\subset\N^2$ and assume that for every $\bn\in I$ a finite section $U_\bn := \mathcal P_{\br(\bn)}\,U\,\mathcal P_\bn$
is given such that
\begin{align}\label{uniboundU}
A\,\zweinorm{\boldsymbol{c}}\le \zweinorm{U_\bn\,\boldsymbol{c}}\quad\text{for all }\boldsymbol{c}\in\ell^2(\Z)
\end{align}
for some constant $A>0$ independent of $\bn$.
Then there are constants $\tilde{c},\tilde{a}>0$, such that for all
$\bn\in I$
\begin{equation}
  \label{eq:c1}
\norm{
U\,(U^*\,U)^{-1} \, e_0
- U_\bn\,(U^*_\bn\,U_\bn)^{-1} \, e_0
	}_2 \le \tilde{c}\,e^{-\tilde{a}\,n_0},
\end{equation}
where $n_0:=\min\{n_1,n_2,r_1(\bn),r_2(\bn)\}$.
\end{theorem}
The proof is deferred to the appendix.

Note that the \emph{row} vector  $\big( U (U^*U)^{-1} ) e_0\big)^* =
e_0^T (U^*U)^{-1} U^* = e_0^T U^\dagger $ is precisely the zeroth row of the  Moore-Penrose pseudoinverse of
$U$ as it arises in  the computation~\eqref{eq:c15}  of the dual windows $\gamma _L$
and $\gamma ^\circ$.
We also note that the decay condition~\eqref{eq:decaying} models the decay of
the entries off a ``ridge'' $\chi (k)$ rather than off-diagonal decay
(in which case $\chi (k) = k$). The above
definition reflects exactly the
behavior of the pre-Gramian matrix $P_g(x)$ of a window with exponential
decay, since $|P_g(x)_{jk}| = |g(x+\alpha j - k/\beta ) | \leq C e^{-
    a\alpha     |j - k/(\alpha \beta ) |}$, in which case $\chi (k) =
  \lfloor k/(\alpha \beta )\rfloor$. Decay conditions of this type occur in wavelet
  theory~\cite{ABK08} and in the theory of Fourier integral operators~\cite{CGNR13}.

We can now prove the main result of our paper, namely that the
numerically computable dual windows $\gamma _L$ converge exponentially
fast to the canonical dual window $\gamma ^\circ $.

\begin{theorem}\label{thm:conv_gammaL}
The dual windows $\gamma_L$, $L\in\N$, in Algorithm~\ref{algo:gamma}
approximate the canonical dual window $\gamma ^\circ $ of $g$ at an exponential rate
\begin{equation}
  \label{eq:c16}
  \zweinorm{\gamma_L-\gamma^\circ} \le \tilde{c}\,e^{-\rho L}.
\end{equation}
\end{theorem}

\begin{IEEEproof}
  We set  $\bn(L) = (\abs{k_1}+L,k_2+L)$ and $\br(L) =
  (\abs{i_1(L)},i_2(L))$ as in Algorithm~\ref{algo:gamma}. The
  matrices $P_L(x)$ in Algorithm~\ref{algo:gamma} are exactly the
  non-symmetric finite sections  $P_L(x) = \mathcal{P}_{\mathbf{r}(L)} P_g(x)
  \mathcal{P}_{\mathbf{n}(L)}$ of the pre-Gramian $P_g(x)$. Then
  Theorem~\ref{thm:bounds-PL} implies that the  finite sections $P_L(x)$
  of the pre-Gramian $P_g(x)$ are left-invertible (on the appropriate
  finite-dimensional subspaces) with constants independent of
  $L$. Therefore   the decay conditions and the uniform bounds in the
  assumptions of
  Theorem~\ref{thm:fs} are fulfilled. Thus for fixed $x$ we obtain
  that
$$
\norm{e_0^T P_g(x)^\dagger - e_0^T P_L(x)^\dagger }_2 \leq \tilde c
e^{-\tilde a n_0(L)} \, .
$$
Finally the approximation of the dual windows $\gamma _L$ and $\gamma
^\circ$ follows from
\begin{align*}
  \norm{\gamma _L - \gamma ^\circ}_2^2 &= \int _{-\infty } ^\infty
  |\gamma _L (x)- \gamma ^\circ (x) |^2 \, dx  \\
&= \int _0 ^\alpha \sum _{j\in \Z } |\gamma _L (x+\alpha j)- \gamma
^\circ (x+\alpha j) |^2 \, dx \\
& = \beta^2 \int _0^\alpha \norm{e_0^T
  P_L(x)^\dagger - e_0^T P_g(x)^\dagger}_2^2 \, dx  \\
&\leq \alpha(\tilde c \beta)^2  e^{-2\tilde a n_0(L)} \, .
\end{align*}
Since  $n_0(L) :=\min\{n_1(L),n_2(L),r_1(L),r_2(L)\} = \min \{ |k_1|+L, k_2
+ L, |i_1(L)|, i_2(L)\} = L + C$ for some integer constant depending
on the window only, the rate of approximation in \eqref{eq:c16} follows.
\end{IEEEproof}
\begin{remark}
(i)
 Theorem~\ref{thm:fs} is not contained in the results on the
 non-symmetric finite section method in \cite{GroeRzeStroh:2010}. The
 selection of rows by $\br(\bn)$ in our assumption (\ref{uniboundU})
 meets only the condition in \cite[Lemma~5.2]{GroeRzeStroh:2010}, which reads as
$$\sup_{\bn\in I}\norm{(U_\bn^*\,U_\bn)^{-1}}_{\ell^2\rightarrow\ell^2}<\infty$$
in our notation. However, the condition in
\cite[Lemma~5.1]{GroeRzeStroh:2010} is not matched and can only be
satisfied  by considerably increasing the number  of rows of $U_\bn$. The
approximations of the canonical dual $\gamma^\circ$ based on the
non-symmetric finite section method in \cite{GroeRzeStroh:2010} do not
provide dual windows, in contrast to our approximations $\gamma_L$.

(ii)  With an analogous proof, we obtain that the dual windows $\gamma_L$ in Algorithm \ref{algo2:gamma} approximate the canonical dual of the EB-spline $B_\Lambda$ at an exponential rate.
\end{remark}

\section{Discretization and implementation}\label{sec:discrete}

For the numerical use of TP functions and EB-splines in signal
analysis it is often  necessary to discretize these windows. We define the
sampling operator $S_\delta$ for a given sampling rate $\delta>0$ by
$$S_\delta g := \sqrt{\delta}\left( g(\delta k) \right)_{k\in\Z}$$
and the periodization operator $P_K$ with period $K>0$ by
$$P_K g(x) := \sum_{k\in\Z} g(x+Kk),\quad x\in[0,K).$$
For discrete signals $\boldsymbol{c}\in\ell^1(\Z)$ we let
$$P_K\,\boldsymbol{c} := \left(\sum_{k\in\Z}c_{j+Kk}\right)_{j=0,\ldots,K-1}\in\C^K.$$
Furthermore we consider the dilation operator
$$D_h g(x) := \sqrt{h}\, g(hx),\quad h>0,$$
which preserves the $L^2$-norm. The exponential decay
\eqref{TP:decay} of TP functions or the compact support of EB-splines
imply  that
$S_\delta\, g$ is in $\ell^1(\Z)\subset\ell^2(\Z)$ and $P_Kg\in L^2([0,K))$.
The combination of  both operators yields  finite discrete signals $P_K S_\delta g\in\C^K$.
For $a,b,M,N,K\in\N$ and $Mb = Na =K$, the discrete Gabor system is defined as
$$\begin{array}{l}
\mathcal G(P_K\,S_\delta\,g,a,\tfrac{1}{M}) =

 \left\{ e^{2\pi i\tfrac{l}{M}\cdot}\,P_K\,S_\delta\,g(\cdot-ka)\middle|
 \begin{array}{l}k=0,\ldots,N-1 \text{ and } \\  l=0,\ldots,M-1 \end{array}\right\}.
 \end{array}
$$
The following result is well known and holds for an extremely general
class of window functions.
All
TP functions of finite type and all EB splines
together with the duals of Algorithms \ref{algo:gamma} and \ref{algo2:gamma}
satisfy this mild condition.

\begin{proposition}\label{prop:Sond}
Let $\alpha,\beta>0$ and
$\alpha\beta =
\tfrac{a}{M} = \tfrac{b}{N}$ and $Mb = Na =K$ with $a,b,M,N,K\in\N$.
Let $g,\gamma\in L^2(\R)$ such that $(g(x+j\alpha))_{j\in\Z}$
and $(\gamma(x+j\alpha))_{j\in\Z}$ are absolutely summable for all $x\in[0,\alpha)$,
and
$$
   \sum_{j\in\Z}\gamma(x+j\alpha)\overline{g(x+j\alpha-k/\beta)} =\beta \delta_{0,k}$$
for all $x\in[0,\alpha)$ and $k\in\Z$.
Then $\mathcal
G(P_K\,S_{\alpha/a}\,g,a,\tfrac{1}{M})$ is a Gabor frame for
$\C^K$ and  $\mathcal G(P_K\,S_{\alpha/a}\,\gamma,a,\tfrac{1}{M})$
is a dual Gabor frame.
\end{proposition}

This statement is proved under slightly different assumptions in \cite{Sond:2007}. The assumptions
in Proposition \ref{prop:Sond} lead directly to the verification of the Wexler-Raz criterion
 in \cite[Theorem A.3]{Sond:2007} for dual Gabor frames of $\C^K$. The details of the proof are omitted here.

\begin{remark}\label{rem:dilation}
Since $S_\delta D_h = S_{\delta h}$, it is helpful to dilate the
function $g$ by the sampling rate. Subsequently we can work with a
sampling rate $\delta = 1$ and consider $P_K S_1 \tilde g =: P_K S
\tilde g$ of some scaled TP function or EB-spline $\tilde
g:=D_{\alpha/a} g$. In many practical situations $a/\alpha$ is
proportional to $\sqrt{K}$.
Thereby the
time-frequency localization of the window is independent of $K$.
\end{remark}

In the remaining part of this section we describe implementations of
discretized TP functions and EB-splines as well as the duals from
Algorithms~\ref{algo:gamma} and~\ref{algo2:gamma} in Section~\ref{sec:known}. For
this purpose, we use some knowledge about the Zak transform of these
functions. For a parameter $\alpha >0$ and a function $f\in L^2(\R)$
with absolutely summable $(f(x+\alpha j))_{j\in\Z}$ for $x\in\R$,
the Zak transform  is defined by
$$\mathrm{Z}_{\alpha}f(x,\omega) := \sum_{j\in\Z}f(x+\alpha j) e^{-2\pi ij\alpha \omega}.$$
The Zak transform is $\alpha$-quasiperiodic in $x$ and $1/\alpha
$-periodic in $\omega$~\cite{janssen-zak88}. For a given periodization parameter $K\in\N$
we obtain the discrete version of a scaled TP function or EB-spline
$\tilde g$ by
\begin{equation}\label{eq:PSg}
P_K S\tilde g(k) = \mathrm Z_K \tilde g(k,0),\quad k=0,\ldots,K-1.
\end{equation}

\subsection{EB-splines}\label{subsec:EB}
Since EB-splines have compact support, their Zak transform is a finite
sum and only requires finitely many point  evaluations of these functions.

\textsl{Case 1}: The EB-spline $B_{\lambda,\ldots,\lambda}$ with  a
single  weight $\lambda\in\R$ of multiplicity $m\in\N$ can be factorized into an exponential and the cardinal polynomial B-spline $N_m$ of order $m$
\begin{align*}
B_{\lambda,\ldots,\lambda} &= e^{\lambda (\cdot)}\chi_{[0,1)} \, \ast \ldots \ast \, e^{\lambda (\cdot)}\chi_{[0,1)} \\
                              &= e^{\lambda (\cdot)}\left( \chi_{[0,1)} \, \ast \ldots \ast \, \chi_{[0,1)}\right) = e^{\lambda (\cdot)}\, N_m.
\end{align*}
Hence it can be evaluated by the well-known algorithm by Cox and
deBoor (see \cite{deBoor:1978}), which is part of the standard signal
processing toolboxes.

\textsl{Case 2}: For EB-splines with pairwise distinct weights $\lambda_1<\cdots<\lambda_m$ and $m\geq2$ Christensen and Massopust \cite{ChrisMass:2010} give the closed form
\begin{equation}\nota
  B_{\lambda_1,\ldots,\lambda_m}(x+k-1)=\sum_{j=1}^m \alpha_j^{(k)} e^{\lambda_j x},~ x\in[0,1),~1\leq k\leq m,
	\end{equation}
with coefficients
\begin{align*}
\alpha_j^{(k)} = \begin{cases}
\prod\limits_{r=1,\atop r\neq j}^{m}(\lambda_m-\lambda_r)^{-1},\ \ &k=1,\\
 \frac{\sum\limits_{1\leq j_1<\cdots<j_{k-1}\leq m,\atop j_1,\cdots,j_{k-1}\neq j}e^{\lambda_{j_1}+\ldots+\lambda_{j_{k-1}}}}{(-1)^{k-1} \,\prod\limits_{r=1,\atop r\neq j}^{m} (\lambda_m-\lambda_r)},\ \ &k=2,\ldots,m.
\end{cases}
\end{align*}

\textsl{Case 3}: For EB-splines with several distinct weights with multiplicities, we use the following four-term recurrence relation, which was stated in \cite{DynRon:1988} and \cite{Ron:1988}.
\begin{theorem}{\bf \cite{DynRon:1988}, \cite{Ron:1988}}
Let $\lambda_1,\ldots,\lambda_N\in\R$, $\lambda_1\neq\lambda_N$ and
$$B_{\lambda_1,\ldots,\lambda_N} =  e^{\lambda_1(\cdot)}\chi_{[0,1)} \, \ast e^{\lambda_2(\cdot)}\chi_{[0,1)} \,
 \ast \ldots \ast \, e^{\lambda_m(\cdot)}\chi_{[0,1)}$$
the associated EB-spline. Then the following recursion holds
\begin{align*}
 & B_{\lambda_1,\ldots,\lambda_N}(x) = \frac{B_{\lambda_1,\ldots,\lambda_{N-1}}(x)-B_{\lambda_2,\ldots,\lambda_N}(x)}{\lambda_1-\lambda_N}

 +\frac{e^{\lambda_1}B_{\lambda_2,\ldots,\lambda_N}(x-1)-e^{\lambda_N}B_{\lambda_1,\ldots,\lambda_{N-1}}(x-1)}
 {\lambda_1-\lambda_N}.
\end{align*}
\end{theorem}

By iteration of this recurrence it is possible to reduce any given
EB-spline into either several lower order EB-splines with pairwise distinct
weights of multiplicity one or only one weight of higher
multiplicity. These can be treated as in the aforementioned cases.

\subsection{TP functions}
In the case of TP functions, we present two different implementations of the computation of $P_KS \tilde g$ in \eqref{eq:PSg}.
For both  we use that TP functions are invariant under dilation.
\begin{lemma}
Let $g$ be a TP function of finite type $N\in\N$ with weights $(\delta_\nu)_{\nu=1}^N$ and $h>0$ a scaling parameter. Then
$$g_h=\sqrt{h}D_h g $$
 is the TP function of finite type in \eqref{eq:TPFFT} with weights $(\tfrac{\delta_\nu}{h})_{\nu=1}^N$.
\end{lemma}
The first implementation uses the identity in \cite[Remark~2]{BanGroeStoe:2013}
$$\mathrm Z_K{g_h}(x,\omega) = [\tfrac{1}{\delta_1},\ldots,\tfrac{1}{\delta_N}\mid r_{x,\omega}]\qquad \forall x\in[0,K),$$
where the right-hand side is the divided difference of $r_{x,\omega}$ with
$$r_{x,\omega}(y) = (-1)^{N-1}h\,\left(\prod_{\nu=1}^N \delta_\nu^{-1}\right)\,
\frac{e^{-hxy}}{1-e^{-K(hy+2\pi i\omega)}}$$
in the knots $\delta_1^{-1},\ldots,\delta_N^{-1}$.

The second  implementation uses the connection of TP functions to
EB-splines. In \cite[Theorem~3.4]{KloStoe:2014} it is shown  that the Zak transform of a TP function can be expressed in terms of the Zak transform of an associated EB-spline.
\begin{theorem}{\bf \cite{KloStoe:2014}}\label{b-splinezak}
Let $g$ be a TP function of finite type with weights $\delta_1,\ldots,\delta_N\in\R$.
With $\lambda_{\nu}:=-\frac{Kh}{\delta_{\nu}}$, $\nu=1,\ldots,N$, we have
$$K\mathrm Z_K g_h(x,0) = \prod_{\nu=1}^N \, \frac{\lambda_\nu}{e^{\lambda_\nu}-1}\,
\mathrm Z_1B_{\lambda_1,\ldots,\lambda_N}(\tfrac{x}{K},0),\ \ x\in[0,1).$$
\end{theorem}
Consequently $P_KS \tilde g$  can be computed by the Zak transform of the corresponding EB-spline as described in \ref{subsec:EB}.

\subsection{Dual windows}\label{imp:duals}
For TP functions $g$ and lattice parameters $\alpha,\beta>0$ with $\alpha\beta<1$, Algorithm~\ref{algo:gamma} in Section~\ref{sec:known} allows us to compute samples $S_{\alpha/N}\gamma_L$, $N\in\N$, of a dual window $\gamma_L$. Likewise, Algorithm \ref{algo2:gamma} provides the sampled dual windows of EB-splines $B_\Lambda$.
Therefore, for a given periodization parameter $K\in\N$ and the time-shift parameter $a\in\N$, we compute
$$P_KS_{\alpha/a}\gamma_L(k)= \sum_{j\in\Z}
(S_{\alpha/a}\gamma_L)(k+jK),\quad k=0,\ldots,K-1 \, .$$
This sum is  finite because of  the compact support of $\gamma_L$.

\begin{example}
We use Algorithm~\ref{algo:gamma} for the computation of the dual Gabor window $\gamma_L$ of the asymmetric TP function $g$ with parameters $\delta=[-1,1,1/3,1/5]$ and lattice parameters $\alpha=2/3$, $\beta=1$. The discretization parameters $K=900$, $a=20$, $b=1/30$, are chosen according to the standard dilation $a/\alpha=\sqrt{K}$ in Remark \ref{rem:dilation}. Figure~\ref{plot:discretedual} shows the discrete TP function $P_K S_{\alpha/a}g$ and its dual window $P_K S_{\alpha/a} \gamma_L$ for $L=20$. The difference $\norm{P_K S_{\alpha/a} (\gamma_L-\gamma^\circ)}_2$ to the discrete canonical dual is $7\cdot 10^{-8}$ measured in the $\ell_2$-norm of $\C^{900}$.
\end{example}

\begin{center}
\begin{figure}
\hspace*{4.5cm
\includegraphics[width=8.8cm]{TPdual.eps}

\caption{Discrete TP function and its dual $\gamma_{20}$ for $a=20$ and $M=30$.}\label{plot:discretedual}
\end{figure}
\end{center}

Sometimes  the frame bounds in the discrete case may  be better than
in the continuous case. Therefore the discrete TP functions or
EB-splines may  even provide Riesz bases at the critical density, as is
explained in the  following recent
result.

\begin{proposition}{\bf \cite{BanGroeStoe:2013},\cite{Klo:2015}}
Let $g$ be a continuous TP function as in \eqref{TPall}, including the infinite type with $\eta = 0$. Assume $\alpha = M\in\N$ and let $\beta = 1/M$ and $K\in\N$ such that $K/M\in\N$.
If $K/M$ is odd, then $\mathcal G(P_KSg,\alpha,\beta)$ is a basis of $\mathbb C^K$.

In addition, assume that $g$ is even, which means that $\{\delta_{\nu}\mid \delta_{\nu}>0\}=\{-\delta_{\nu}\mid \delta_{\nu}<0\}$.
If $M$ is odd, then $\mathcal G(P_KSg,\alpha,\beta)$ is a basis of $\mathbb C^K$.
\end{proposition}

\begin{remark}
In the critical case $\alpha \beta =1$ the computation of the dual window $\gamma$
cannot be performed by the aforementioned discretization procedure, as
the Gabor system $\gab$ is not a frame in $L^2(\R)$. In this case,
 the usual method using the discrete Fourier transform and the
discrete Zak transform of $P_K S\tilde g$ should be applied for the
computation of the discrete dual window~\cite{BoelHlaw97}.
\end{remark}

\section*{Acknowledgement}
First discussions on the topic of this article were performed while J. St\"ockler visited the Erwin Schr\"odinger Institute. The authors are grateful to Z.~{P}r\r{u}\v{s}a and P.~L. {S}{\o}ndergaard for many helpful discussions concerning the implementation of the algorithms for the computation of the canonical dual window which were held at the Strobl conference on Modern Time-Frequency Analysis in 2014.

\section*{Appendix}
In the appendix we provide the technical details of the proofs of the
main theorems. The proof of Theorem~\ref{thm:bounds-PL} depends heavily on
\cite[Theorem~9]{GroeStoe:2013}.

\begin{IEEEproof}[Proof of Theorem~\ref{thm:bounds-PL}]
The pre-Gramian is bounded on $\ell ^2(\Z )$ as a consequence of the
exponential decay of $g$ and Schur's test. Therefore the finite
sections $P_L(x) = \mathcal P_{\br(\bn)}\,P_g(x) \,\mathcal P_\bn$ are
uniformly bounded.

To prove the existence of a lower bound $A$, we construct a left
inverse of $P_L(x)$  by adapting the proof of
\cite[Theorem~9]{GroeStoe:2013}.
By step~2 of Algorithm~\ref{algo:gamma}  $P_L(x)$ has columns indexed by $k,
k_1-L\le k\le k_2+L$ and rows indexed by $j, i_1(L)\le j\le i_2(L)$,
thus every  left-inverse  $Q_L(x)$ has columns indexed by $i_1(L)\le
j\le i_2(L)$ and rows indexed by $k_1-L\le k\le k_2+L$. We will
construct these  rows one by one in three steps.
\medskip

\noindent {\emph Step 1.} For every index $-L\le \ell \le L$, we choose the following submatrix $R_\ell$
of $P_L(x)$. Given $x$ and $\ell \in \mathbb{Z}$, we choose  $y_\ell\in [0,\alpha)$ and $j_\ell\in\Z$, such that
\[
    y_\ell = x+j_\ell\alpha-\frac{\ell}{\beta}.
\]
Then we define the matrix $R_\ell=P_0(y_\ell)$ by using the
same definitions as in step~3 of Algorithm \ref{algo:gamma} with $L=0$;
i.e., we let
\begin{align*}
     i_1^\ell(0) &= \left\lfloor \frac{k_1 + m -
        1}{\alpha\beta}-\frac{y_\ell}{\alpha}\right\rfloor+1,\\
     i_2^\ell(0) &=
    \left\lceil \frac{k_2 - n +
        1}{\alpha\beta}-\frac{y_\ell}{\alpha}\right\rceil-1,
 \end{align*}
and
\[
   R_\ell=\left( r_{\ell;j,k}\right)
	_{i_1^\ell(0)\le j\le i_2^\ell(0),~k_1\le k\le k_2},
\]
with $r_{\ell;j,k}=g\left(y_\ell+j\alpha-\frac{k}{\beta}\right)$. Note that
\[
   y_\ell+j\alpha-\frac{k}{\beta} = x+(j+j_\ell)\alpha-\frac{\ell + k}{\beta}.
\]
Hence,   the columns of $R_\ell$, as indexed by $k_1\le k\le k_2$, correspond to sections of
 columns of $P_L(x)$ indexed by $\ell+k_1\le k\le \ell+k_2$.
 More precisely, $R_{-L}$ contains sections of the first
$k_2-k_1+1$ columns of $P_L(x)$;  with increasing $\ell$ the selection
of columns is shifted to the right;  and $R_L$ contains sections of the
last $k_2-k_1+1$ columns of $P_L(x)$. For the rows of $R_\ell$, we
observe that
\[
   i_1(L)\le \left\lfloor \frac{k_1 + m -
        1}{\alpha\beta}-\frac{x}{\alpha}+ \frac{\ell}{\alpha\beta}\right\rfloor+1
        =i_1^\ell(0)+j_\ell
\]
and $i_2(L)\ge i_2^\ell(0)+j_\ell$. To summarize, each $R_\ell $  is a $(i_2^\ell (0) -
i_1^\ell (0) +1) \times (k_2 - k_1+1)$-submatrix of $P_L(x)$ with a
dimension independent of  $|\ell | \leq L$.
\medskip

\noindent{\emph Step 2.} The arguments in \cite[Theorem~8]{GroeStoe:2013}
relate left-inverses of $R_\ell=P_0(y_\ell)$ to left-inverses
of the biinfinite matrix $P_g(y_\ell)$. It is shown that
\begin{itemize}
\item $R_\ell$ has full column rank,
\item there exists a uniform bound $C>0$, which does not
depend on  $y_\ell\in[0,\alpha)$, and left-inverses
$\Gamma_\ell$ of $R_\ell$ such that
\[
   \| \Gamma_\ell\| \le C\quad\mbox{for all}\quad
	-L\le \ell\le L,
\]
\item the rows with index $k_1\le k\le 0$ of $\Gamma_\ell$
are orthogonal to all columns with index $k'>k_2-n$ of the biinfinite matrix $P_g(y_\ell)$, and hence orthogonal to all columns $k'>\ell+k_2-n$ of $P_L(x)$.
Likewise, rows $0\le k\le k_2$ of $\Gamma_\ell$
are orthogonal to all columns with index $k'<\ell+k_1+m$ of $P_L(x)$.
\end{itemize}

\noindent{\emph Step 3.}
With these properties, we obtain the left-inverse $Q_L(x)$ of $P_L(x)$ by
defining the rows of $Q_L(x)$ as follows:
\begin{itemize}
\item We start with $\Gamma_{-L}$ and take its rows $k_1\le k\le 0$ as the first
 rows of $Q_L(x)$, extended by zeros such that
\begin{align*}
     &(Q_L(x))_{k-L,j+j_{-L}}

     = \begin{cases} (\Gamma_{-L})_{k,j},&~  i_1^{-L}(0)\le
		j\le i_2^{-L}(0),\\
		0,&~\mbox{otherwise}.
		\end{cases}
\end{align*}
\item For $-L+1\le \ell\le L-1$ we take the row
with index $0$ of $\Gamma_\ell$ and extend this row by
zeroes,
\[
     (Q_L(x))_{\ell,j+j_\ell}= \begin{cases} (\Gamma_{\ell})_{0,j},&~ i_1^\ell(0)\le
		j\le i_2^\ell(0),\\
		0,&~\mbox{otherwise}.
		\end{cases}
\]
\item We end with $\Gamma_{L}$ and take its rows $0\le k\le k_2$ as the last
 rows of $Q_L(x)$, extended by zeros such that
\begin{align*}
     &(Q_L(x))_{k+L,j+j_{L}}

     = \begin{cases} (\Gamma_{L})_{k,j},&~
     i_1^L(0)\le j\le i_2^L(0),\\
		0,&~\mbox{otherwise}.
		\end{cases}
\end{align*}
\end{itemize}
It is clear that all entries of $Q_L(x)$ are bounded  by
the constant $C$ in step 2. Moreover,
every row and column of $Q_L(x)$ has at most
\[
  \max_{-L\le \ell\le L}  (i_2^\ell(0)-i_1^\ell(0)+1 ) \le \frac{k_2-k_1+1}{\alpha\beta}
\]
nonzero entries. Therefore, by Schur's test, we obtain that
\[
  \|Q_L(x)\| \le C\,\frac{k_2-k_1+1}{\alpha\beta}.
\]
Since $Q_L $ is a left inverse of $P_L$, we obtain
$$
\norm{c}_2 = \norm{Q_L(x) P_L(x) c}_2 \leq C \,
\frac{k_2-k_1+1}{\alpha\beta}  \norm{P_L(x)c}_2 ,
$$
and we may choose $A = (C \,
\frac{k_2-k_1+1}{\alpha\beta} )^{-1} $ as a lower bound in~\eqref{PLbounds}.
This completes the proof of (\ref{PLbounds}). We conclude that
$\zweinorm{e_0^T\,P_L(x)^\dagger}\leq C \,
\frac{k_2-k_1+1}{\alpha\beta} =  A^{-1}$. Therefore
\begin{align*}
  \zweinorm{\gamma_L}^2 &= \int _{-\infty } ^\infty |\gamma _L(x)|^2
  \, dx \\
& = \beta^2\, \int_0^\alpha \sum _{j\in \Z } |\gamma _L(x+j\alpha )|^2 \,dx \\
&= \beta ^2 \int _0^\alpha  \zweinorm{e_0^T\,P_L(x)^\dagger}^2\, dx \leq \alpha\,\beta^2\,A^{-2}.
\end{align*}
\end{IEEEproof}

In the following we will need a well known result of
Jaffard~\cite{Jaff:1990} and Baskakov~\cite{Bas90}.

\begin{proposition} \label{jaffexp}
  Let $A = (A_{jk})_{j,k \in \Z } $ be a biinfinite matrix with
  exponential off-diagonal decay, i.e., there exist  constants $C,a>0$, such that
  \begin{equation}
    \label{eq:c89}
|A_{jk}| \leq C e^{-a |j-k|} \quad \forall j,k \in \Z \, .
  \end{equation}
If $A$ is invertible, then $A^{-1} $ also possesses exponential
off-diagonal  decay
and there exist $C'> 0$ and $0< \tilde a < a$, such that
$$
|(A^{-1} )_{jk}| \leq C' e^{-\tilde a |j-k|} \qquad \forall j,k\in \Z
\, .
$$
\end{proposition}

\begin{corollary} \label{jafffsm}
  Let $A$ be  positive and invertible on $\ell ^2(\Z )$ with
  exponential off-diagonal decay~\eqref{eq:c89} and let $\mathcal{P}_{\bn }A \mathcal{P}_{\bn } $ be a
  sequence of finite sections. Then every matrix  $\mathcal{P}_{\bn }A
  \mathcal{P}_{\bn }$ is invertible on $\mathcal{P}_{\bn}\ell ^2(\Z)$
  and there exist $C'>0$ and $\tilde a,
  0<\tilde a <a$, such that
  \begin{equation}
    \label{eq:c17}
|(\mathcal{P}_{\bn }A  \mathcal{P}_{\bn })^{-1}_{jk}|\leq C' e^{-\tilde a
  |j-k|} \qquad  \text{ for } -n_1 \leq j,k \leq n_2 \, .
  \end{equation}
\end{corollary}
\begin{IEEEproof}
  The proof follows~\cite{HagRochSilb:2001} and \cite{GroeRzeStroh:2010}. Let $\mathcal{A}$ be the matrix obtained
  by stacking the finite sections $\mathcal{P}_{\bn }A
  \mathcal{P}_{\bn }$ along the diagonal. Then $\mathcal{A}$ possesses
  exponential off-diagonal decay~\eqref{eq:c89}. Since $A$ is invertible and positive, its  spectrum
  $\sigma (A)$ is
  contained in an interval $[A,B]$ for some $A,B>0$, consequently  the spectrum of the
  restriction of $\mathcal P_\bn\,A\,\mathcal P_\bn$ on
  $\mathcal P_\bn\ell^2(\Z)$ is also contained in $[A,B]$ and every
  finite section is invertible  on $\mathcal P_\bn \ell ^2(\Z
  )$. Therefore the stacked matrix $\mathcal{A}$ is invertible on
  $ \oplus \mathcal P_\bn\ell^2(\Z) \simeq \ell ^2(\Z )   $. By
  Proposition~\ref{jaffexp} $\mathcal{A}^{-1} $ possesses exponential
off-diagonal   decay. Since $\mathcal{A}^{-1} $ consists of the blocks
  $(\mathcal{P}_{\bn }A  \mathcal{P}_{\bn })^{-1}$, they satisfy
$|(\mathcal{P}_{\bn }A  \mathcal{P}_{\bn })^{-1}_{jk}|\leq C' e^{-\tilde a
  |j-k|}$ for $ -n_1 \leq j,k \leq n_2 $.
\end{IEEEproof}
We remark that clearly every finite matrix possesses exponential
off-diagonal decay. The point is that the constants may be chosen
independently of the size of the finite section.

\begin{IEEEproof}[Proof of Theorem~\ref{thm:fs}]
Recall that, for   $\boldsymbol{n} =
(n_1,n_2) \in\N^2$, $\mathcal P_{\boldsymbol{n}}\,b =
(\ldots,0,b_{-n_1},b_{-n_1+1},\ldots,b_{n_2-1},b_{n_2},0,\ldots)^T$   is the orthogonal projection onto
$\mathcal P_{\boldsymbol{n}}\,\ell^2(\Z) \cong \C^{n_1+n_2+1}$. We
write  $U_\bn= \mathcal
P_\br\,U\,\mathcal P_\bn$ for a non-symmetric finite section of
$U$.  In the assumption of Theorem~\ref{thm:fs} $\br $ depends on
$\bn$, but we will omit this dependence in the notation. All operator norms are in $\ell^2(\Z)$.

Let $b:= U\,(U^*\,U)^{-1} \, e_0$ and $d_\bn := U_\bn\,(U^*_\bn\,U_\bn)^{-1}
\, e_0$. We decompose the norm into three parts
\begin{align}\notag
\zweinorm{b - d_\bn} &\le \zweinorm{(U - U\,\mathcal P_\bn)\,(U^*\,U)^{-1}\,e_0}\\ \label{pf:finsec}
+& \zweinorm{U\,\mathcal P_\bn\,\Big((U^*\,U)^{-1} - (\mathcal P_\bn\,U^*\,U\,\mathcal P_\bn)^{-1}\Big)\,e_0}\\ \notag
+& \zweinorm{U\,\mathcal P_\bn\,((\mathcal P_\bn\,U^*\,U\,\mathcal P_\bn)^{-1} - U_\bn\,(U^*_\bn\,U_\bn)^{-1})\, e_0}.
\end{align}
\medskip

\noindent{\emph 1.} Note that $U^*\,U$ is invertible and positive. Since $U$
  fulfills the decay property (\ref{eq:decaying}), it is easy to see
  that  the symmetric
  matrix $U^*\,U$ decays exponentially off the diagonal, i.e., for
  some constants $C,a>0$
$$
|(U^*U)_{jk}| \leq C e^{-a |j-k|} \qquad j,k \in \Z \, .
$$
By Proposition~\ref{jaffexp}
the inverse matrix inherits the exponential decay, and thus  there exist  constants $C'
>0$  and   $\tilde a , 0<  \tilde a < a$,

  such that $|(U^*\,U)^{-1}_{jk}|  \leq   C' e^{-\tilde a |j-k|} $ for all $j,k \in \Z $.
Hence the entries of the vector $v:=(U^*\,U)^{-1}\,e_0$ also decay
exponentially as
\begin{equation}\label{vj_decay}
\abs{v_j} \le c_1\,e^{-\tilde{a}\,\abs{j}}  \quad j\in \Z \,.
\end{equation}
Since
\begin{align*}
\zweinorm{(I-\mathcal{P}_\bn ) v}^2 &= \sum _{j= -\infty } ^{-n_1-1}
|v_j|^2 + \sum _{j= n_2+1 } ^{\infty} |v_j|^2
\leq C' \sum _{|j|> n_0} e^{-2\tilde a |j|} = \mathcal{O}(e^{-2\tilde a
  |j|})\, ,
\end{align*}
the decay property \eqref{vj_decay} implies that
\begin{align*}
\zweinorm{(U - U\,\mathcal P_\bn)\,(U^*\,U)^{-1}\,e_0} &\le
\norm{U}\,\zweinorm{(I - \mathcal P_\bn)\,v}
\le
c_2\,e^{-\tilde{a}\,n_0}.
\end{align*}
\medskip

\noindent{\emph 2.}  Since $U^*\, U$ is invertible and positive with   spectrum
  $\sigma (U^*U) \subseteq [A,B]$ for $A,B>0$,   the spectrum of the
  finite sections  $\mathcal P_\bn\,U^*\,U\,\mathcal P_\bn$ on
  ${\mathcal P_\bn\ell^2(\Z)}$ is also contained in $[A,B]$.  As in the finite section method in \cite{HagRochSilb:2001} and with $v=(U^*\,U)^{-1}\,e_0$, we obtain
\begin{align*}
&\zweinorm{((U^*\,U)^{-1} - (\mathcal P_\bn\,U^*\,U\,\mathcal P_\bn)^{-1})\,e_0}
=\zweinorm{(\mathcal P_\bn\,U^*\,U\,\mathcal P_\bn)^{-1}\,\mathcal P_\bn\,U^*\,U\,(\mathcal P_\bn-I)\,v}
\le \norm{(U^*\,U)^{-1}}\,\norm{U^*\,U}\,\zweinorm{(\mathcal P_\bn - I)\,v}
\end{align*}
and consequently
\begin{align*}
\zweinorm{U\,\mathcal P_\bn\,\big((U^*\,U)^{-1} - (\mathcal
  P_\bn\,U^*\,U\,\mathcal P_\bn)^{-1}\big)\,e_0}&

  \le c_3\,e^{-\tilde{a}\,n_0}&.
\end{align*}
\medskip

\noindent{\emph 3.} To treat the third term in~\eqref{pf:finsec},   we need a
  geometric interpretation of the rows of the Moore-Penrose
  pseudoinverse. Let
$$b_\bn := (U\,\mathcal P_\bn)\,(\mathcal P_\bn\,U^*\,U\,\mathcal P_\bn)^{-1}\,e_0$$
and $d_\bn$ as above. Then $b_\bn$ is  the transpose of the zeroth row
of the Moore-Penrose pseudoinverse of $U\,\mathcal P_\bn$. By
Corollary~\ref{jafffsm} $(\mathcal P_\bn\,U^*\,U\,\mathcal
P_\bn)^{-1}$ satisfies \eqref{eq:c17} independently of $\bn$. Therefore
the same argument as in the first part implies  the decay property
\begin{align}\label{dec:b_n}
\abs{(b_\bn)_j} \le c_4\,e^{-\tilde{a}\,\abs{j}}.
\end{align}
It is
essential  that the constants are independent of $\bn$.
Since the Moore-Penrose pseudoinverse of $U\,\mathcal P_\bn$ is also a
left-inverse, we have
$$
b_\bn\,\bot\,V_\bn := \mathrm{span}\{u_k\mid -n_1\le k\le n_2,\ k\neq 0\} ,
$$
where $u_k$, $k\in\Z$, are the columns of the matrix $U$.
Likewise for $U_\bn = \mathcal{P}_{\br} U \mathcal{P}_\bn $ we have
\begin{equation*}
d_\bn\,\bot\,\mathcal P_\br V_\bn\, .
\end{equation*}

Using this orthogonality, we rewrite the vectors $b_\bn $ and $d_\bn $
as follows. Let $\Pi_W$ denote  the orthogonal projection onto some
subspace $W$. Now  set
\begin{equation*}
  \tilde{b}_\bn := (I - \Pi_{V_\bn})\, u_0\quad \text{ and }\quad
  \tilde{d}_\bn := (I - \Pi_{\mathcal P_\br V_\bn})\, \mathcal
  P_\br\,u_0 \, .
\end{equation*}
Since $b_\bn \in \mathrm{Im} (U \mathcal{P}_\bn )$ and $d_\bn \in
\mathrm{Im} (U_\bn )$, we obtain
\begin{equation}\label{normal}
b_\bn = \frac{\tilde{b}_\bn}{\norm{\tilde{b}_\bn}_2^2} \quad \text{
  and } \quad d_\bn =
\frac{\tilde{d}_\bn}{\norm{\tilde{d}_\bn}_2^2}\, .
\end{equation}
 The normalization in (\ref{normal}) is obtained from
\begin{align*}
&\sprod{b_\bn}{u_0} = 1 \quad\text{and}
\quad \sprod{\tilde{b}_\bn}{u_0} = \sprod{\tilde{b}_\bn}{\tilde{b}_\bn + \Pi_{V_\bn}\,u_0} = \norm{\tilde{b}_\bn}_2^2,\\
&\sprod{d_\bn}{\mathcal P_\br u_0} = 1 \quad\text{and}
\quad \sprod{\tilde{d}_\bn}{\mathcal P_\br u_0} = \sprod{\tilde{d}_\bn}{\tilde{d}_\bn + \Pi_{\mathcal P_\br V_\bn}\,\mathcal P_\br u_0} = \norm{\tilde{d}_\bn}_2^2.
\end{align*}
As the third term   in (\ref{pf:finsec}) equals $\norm{b_\bn-d_\bn}_2$, we
first consider $\norm{\tilde{b}_\bn - \tilde{d}_\bn}_2$. For this
purpose we write
\begin{align} \label{hm1}
\tilde{b}_\bn - \tilde{d}_\bn &= (I-\mathcal P_\br)\,(u_0
-\Pi_{V_\bn}\,u_0) + \Pi_{\mathcal P_\br V_\bn}\,\mathcal P_\br\,u_0
- \mathcal P_\br\,\Pi_{V_\bn}\,u_0 \notag \\
                              &= (I-\mathcal P_\br)\,\tilde{b}_\bn +
                              \Pi_{\mathcal P_\br V_\bn}\,\mathcal
                              P_\br\,(u_0 - \Pi_{V_\bn}\,u_0) \notag\\
                              &=    (I-\mathcal P_\br)\,\tilde{b}_\bn
                              +  \Pi_{\mathcal P_\br
                                V_\bn}\,\mathcal P_\br\,\tilde{b}_\bn.
\end{align}
By the assumption (\ref{uniboundU}), the truncated columns $\mathcal
P_\br\,u_k$, with $-n_1\le k\le n_2$ and $k\neq 0$, form a Riesz basis
for $\mathcal P_\br V_\bn$ with lower Riesz bound $A$. Furthermore
it holds that
\begin{align*}
\zweinorm{\Pi_{\mathcal P_\br V_\bn}\,\mathcal P_\br\,\tilde{b}_\bn}^2
&\le A^{-2}\, \sum_{-n_1\le k\le n_2,\,k\neq 0}\abs{\sprod{\Pi_{\mathcal P_\br V_\bn}\,\mathcal P_\br\,\tilde{b}_\bn}{\mathcal P_\br\,u_k}}^2\\
&= A^{-2}\, \sum_{-n_1\le k\le n_2,\,k\neq 0}\abs{\sprod{\mathcal P_\br\,\tilde{b}_\bn}{u_k}}^2\\
&= A^{-2}\, \sum_{-n_1\le k\le n_2,\,k\neq 0}\abs{\sprod{(\mathcal P_\br-I)\,\tilde{b}_\bn}{u_k}}^2\\
&\le A^{-2}B\, \zweinorm{(\mathcal P_\br-I)\,\tilde{b}_\bn}^2,
\end{align*}
where $B=\norm{U}^2$ denotes the Bessel bound of all columns $u_k$,
$k\in\Z$.
Taking the $\ell ^2$-norm in \eqref{hm1} and substituting the above
estimate, we obtain
$$\norm{\tilde{b}_\bn - \tilde{d}_\bn}_2 \le c_5\,\norm{(I-\mathcal P_\br)\,\tilde{b}_\bn}_2.$$
We now return to $\norm{b_\bn-d_\bn}_2$. It is an easy exercise that
for nonzero vectors $y,w$ we have
$$\zweinorm{\frac{y}{\zweinorm{y}^2} - \frac{w}{\zweinorm{w}^2}} \le \frac{3\,\zweinorm{y-w}}{\min\{\zweinorm{y}^2,\zweinorm{w}^2\}}.$$
Using once more that $b_\bn \in \mathrm{Im} (U \mathcal{P}_\bn )$ and $d_\bn \in
\mathrm{Im} (U_\bn )$,  the assumption (\ref{uniboundU}) implies
that
$$0<A\le\norm{\tilde{b}_\bn}_2,
\norm{\tilde{d}_\bn}_2\le\zweinorm{u_0} \, .$$
 By the decay property (\ref{dec:b_n}) we obtain
\begin{align*}
\norm{b_\bn - d_\bn}_2\le\tfrac{3}{A^2}\, \norm{\tilde{b}_\bn - \tilde{d}_\bn}_2&\le\tfrac{3\,c_5}{A^2}\, \norm{(I-\mathcal P_\br)\,\tilde{b}_\bn}_2
\le c_6\,e^{-\tilde{a}\,n_0}.
\end{align*}
To finish the proof of Theorem~\ref{thm:fs},  we add the three contributions in \eqref{pf:finsec} and obtain the
convergence rate \eqref{eq:c1} with a constant  $\tilde{c} = c_2+c_3+c_6$.
\end{IEEEproof}

redpajama_set_name: RedPajamaArXiv


text:
Russian ship Admiral Gorshkov performs hypersonic missile launch
Into the Deep with Unmanned Undersea Vehicles
Recent incidents under the sea could place new importance on the development and role of unmanned vehicles. Richard B Gasparre investigates.
In spring 2007, a US-based salvage company located the wreck of a Napoleonic-era Spanish treasure ship. The company, Odyssey Marine Explorations, asserted that the wreck was in international waters, but Spain, seeking sunken gold and silver worth up to $500m, took legal steps to contest ownership and recovery – and essentially blockaded Odyssey's main salvage vessel in Gibraltar by threatening to seize it if it came out.
Leading Guide to Submarine and Submersible Suppliers for the Naval Industry
The document includes detailed information on the manufacturers and suppliers and their products, along with contact details, to inform your purchasing decision.
By Naval Technology Buyers Guide
By clicking the Download Free Buyer's Guide button, you accept the terms and conditions and acknowledge that your data will be used as described in the Naval Technology Buyers Guide privacy policy
By downloading this Buyer's Guide, you acknowledge that we may share your information with our white paper partners/sponsors who may contact you directly with information on their products and services.
Thank you.Please check your email to download the Buyer's Guide.
SUBSEA ACTIVITY
That same summer, Russian scientists and legislators travelled to the North Pole on an icebreaker and two submarines. They cut through the ice pack to dive more than two miles to the bottom of the ocean. On the seabed, they planted the Russian flag – and after surfacing, asserted that the expedition had strengthened Moscow's claims to nearly half the Arctic seabed. Needless to say, Canada, Norway, Denmark and most of the UN community were not amused.
In early February of 2008, many nations along the Indian Ocean littoral, including most Persian Gulf states, experienced severe internet disruptions that crippled global business operations for two weeks. The disruptions stemmed from damage to two pairs of undersea cables that apparently represented a key portion of the Near East's internet backbone.
Subsequent investigations determined that two freighters cut two pairs of cables, one off Egypt and one off the UAE, about a week apart. Though quite coincidental, the accidents were apparently due to negligence rather than malice, although hyperalert observers emphasised that one ship was Iraqi and the other 'Korean' (presumably South Korean – if it had been North Korean, there doubtless would have been more sabre rattling).
This may all sound like a hook for a new paperback techno-thriller, but there's a serious question here: What do these events have in common?
DEEPWATER ASSETS
The answer, of course, is that all three events involve deepwater seabed assets. And based on the traditional notion (most recently mentioned in Enforcing Fair Trade by Dr Gareth Evans – see link below) that navies follow trade the way litigators follow the money, the implication is that deep-ocean and seafloor resources of various kinds will require increased protection in the future.
In particular, this year's cable-cutting incidents give the relatively new paradigm of information warfare an even newer twist. If two unwitting captains could cripple Persian Gulf communications for a week, what would happen if disgruntled or corrupt officers – or more likely teams of dedicated pirates or saboteurs – were actually trying to cause damage? More insidiously, what would happen if China sent a covert team of divers and hackers to intercept these data streams – or to introduce viruses into them?
"Seaborne terrorists could use merchant ships or small coastal boats for cover and transportation."
Protecting assets on the seabed is analogous to defending against insurgent attacks on land. Just as guerrillas can blend into the population and wait for a moment of maximum advantage, seaborne terrorists could use merchant ships or small coastal boats for cover and transportation. Heretofore, the defensive advantage of seagoing targets has been the relative lack of traffic and cover at sea – but if insurgents can take the offensive underwater, it's a whole different ball game, as anti-submarine specialists will tell you.
UNDERWATER WARFARE
Granted, building combat-ready submersibles is not as easy as playing with radio-controlled toy sailboats; in fact, none of the likely suspects have even built their own autonomous torpedo yet (as far as we know).
The critical challenge for underwater operations, especially at a distance, is adaptive communications and control, as traditional electro-optical methods don't work underwater. As a result, most torpedoes generally trail a wire spool back to their launching vessel, so as to utilise the vastly more powerful shipboard sensor and computer resources.
Nevertheless, civilian submersibles for salvage, cable-laying, oil drilling, and similar operations have been in place for decades now, so a dedicated saboteur wouldn't have to be the next Da Vinci to effectively modify existing technology. Moreover, using traditional naval vessels (and especially highly in-demand submarines) to guard such resources would be a losing proposition by every doctrinal and logistical consideration.
However, the problem suggests its own solution: using unmanned undersea vehicles (UUVs) for perimeter defence. The consensus among designers and users on land is that autonomous systems, whether mobile or fixed, provide the greatest benefit when used for operations that are '3D' – dull, dirty or dangerous. At first sight, using UUVs for guard work is no different:
"Deep-ocean and seafloor resources of various kinds will require increased protection in the future."
Sentry duty, wherever it occurs, is dull
Operating on the ocean floor can be just as dirty as operating in a swamp
Any location that presents the risk of pressure crush and drowning ought to be regarded as dangerous
The US Navy has already deployed UUVs for mine countermeasure operations. In particular, Boeing's Long-term Mine Reconnaissance System can launch from, and return to, underway submerged submarines, as was demonstrated in November 2007.
Assuming some kind of mother ship / robotic swarm paradigm, the key developmental question is now whether UUVs can be operated at long ranges and for extended durations, whether by cable (seems unlikely), satellite (possible) or other innovative or revolutionary means (synthetic whale song).
At any rate, one thing is clear – as the stakes rise, so will the pressure to find a solution.
Unmanned Underwater Vehicles (Defense) - Thematic Research
Lithium-Ion Batteries for Underwater Applications - Thematic Research
Disruptor Profile: Natural Machines
Cryptocurrencies - Thematic Research
Unmanned Ground Vehicles (Defense) - Thematic Research
Boeing China Inc
Russo-Ukraine war equipment loss ten times that of Moscow's Chechen conflicts

redpajama_set_name: RedPajamaCommonCrawl


The China magnetic separating production line usually completes three grade separations of the mineral powder and finally realizes the wet material magnetic separation.
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Zenith Mining Company adopts the stagewet grinding magnetic separation process, adopting the discarding the tailing during wet magnetic separation. » Learn More.


Ernst Urbainczyk joined the team about 20 years ago and is CEO of Gitta Sells & Associates International. His functions have evolved over the years and he is now mainly focused on the company's future success and growth. His background as a hotel executive took him around the world where he worked in such countries as Germany, France, England, Switzerland, Sweden, and Bermuda.


The Sun City Hotel & Casino was the first hotel built on the resort and is often referred to as the Main Hotel. The Hotel is central to the main casino and theatre with 340 pool and lake facing rooms on the edge of the world-renowned Gary Player Golf Course.
Sun City has no shortage of activities for the whole family from the two championship golf courses designed by the legendary Gary Player to the Valley of Waves, Sun City's world-class water park for the entire family to enjoy.
The Resort additionally offers a casino, shopping, family & kids entertainment as well as safari and wildlife in the nearby Pilanesberg National Park.
The Sun City Hotel has 340 lagoon and lake facing rooms and 6 luxury suites and 2 presidential suites. A free shuttle passes by every 15 minutes, taking guests to the Valley of Waves, Entertainment Centre and other activities within the resort.
The Sun City Hotel is 4-star Sun City Accommodation and is ideal for delegates attending conferences at the resort or for guests who want to be in the think of the action.


text:
Jim Geoghan est un producteur et scénariste américain.

Il est surtout connu pour être le co-créateur et le producteur exécutif de la série La Vie de palace de Zack et Cody, pour laquelle il a été nommé deux fois aux Primetime Emmy Awards dans la catégorie « Meilleur programme pour enfants », en 2007 et 2008.

Biographie

Jim Geoghan a étudié à l'Institut de technologie de New York, où il obtient un Bachelor of Fine Arts en 1969. Il fait ensuite du stand-up, notamment en rejoignant le trio « Divided We Stand » en 1973.

Il a participé, en tant que producteur et scénariste, à de nombreuses séries télévisées célèbres, telles que Ricky ou la Belle Vie, , Drôle de vie ou encore La Vie de famille. Son expérience dans les programmes familiaux lui permet de travailler avec Disney Channel,  pour laquelle il produit La Vie de palace de Zack et Cody. Cette série obtient un grand succès, et Jim Geoghan est nommé deux fois aux Primetime Emmy Awards dans la catégorie « Meilleur programme pour enfants », en 2007 et 2008.

Jim Geoghan écrit aussi des pièces de théâtre, notamment Only Kidding qui a connu 500 représentations à Broadway. Saluée par la critique, cette pièce a été nommée pour deux Drama Desk Awards.

Vie privée
Jim Geoghan vit à Los Angeles (Californie) avec son épouse, l'actrice Annie Gagen, et leur fille, Genevieve.

Filmographie

Références

Lien externe

Scénariste américain
Producteur américain de télévision
Date de naissance non renseignée (XXe siècle)
Lieu de naissance inconnu

redpajama_set_name: RedPajamaWikipedia


text:
Whalers kill 3 more orcas in St. Vincent (+video)
16 Apr 2018 - 11:32PM
Whalers process three orcas in Barrouallie on Monday, one day after they were killed. (iWN photo)
By Kenton X. Chance
The whaler whose crew harpooned and killed three orcas off Barrouallie, a town in western St. Vincent, on Sunday, has defended his action, saying that the kill is important to the livelihood of townsmen, and that they have broken no local law.
Three orcas, one male and two females, measuring 30 and 24 feet, were caught as they scattered a pod of short-finned pilot whales that the whalers were initially pursuing, the captain of the crew, Samuel Hazelwood, told iWitness News on Monday.
Hazelwood, who has been whaling for more than 30 years, spoke to iWitness News amidst a group of hostile whaling industry workers who were carving up the carcasses for resale to vendors.
https://youtu.be/cH0XJU7XbHg
He was mindful of the negative press that comes with the killing of orcas, but noted that the government was yet to pass a law banning the killing of orcas, as Prime Minister Ralph Gonsalves said in April 2016 that his government would have done.
"The prime minister said he is going to ban them (the killing of orcas). I have been fishing for 30-something years now and I came and met the older fishermen catching the killer whales. They say it is endangered [but] when you go and research on YouTube and Facebook, it is not an endangered species," Hazelwood told iWitness News.
In April 2016, amidst local and international outcry after the whalers harpooned and killed two orcas in the presence of cruise visitors on a whale and dolphin watch tour, Gonsalves had said that his government would pass a law to ban the killing of orcas.
Hazelwood said that Barrouallie whalers would obey such a law if it were passed.
"The prime minister said he is going to ban us. Whenever he put the ban in place like the [ban on the catching of ] turtle and they say stop catch them (orcas), we [will] adhere. But there is no law right now to say don't catch them. This is something that we have been doing for centuries," Hazelwood told iWitness News.
"Whenever they put the ban," we don't have any problems.
He said that on Sunday, his brother called him saying that he was pursuing a pod of pilot whales.
"And when we were chasing the black fish, the killer whales came and started running them down [so] we couldn't catch any. When you see them (orcas), life is really difficult for us."
Hazelwood accused the prime minister of not understanding the plight of Barrouallie whalers and those who depend on the industry for their livelihood.
"This is something that a lot of people in Barrouallie [depend on]. These killer whales that we catch here today, you are going to get more than a hundred and something people making a money for the day. All of the shops are going to benefit, so it is something that we make good money from."
Whaler Samuel Hazelwood has defended his actions, saying he has broken no law and is ensuring his own livelihood. (iWN photo)
Hazelwood said that the only parts of the whales that are thrown away are the bones and the intestines.
"Everything [else] is sold: teeth, everything," he said.
In St. Vincent and the Grenadines, the meat of the whale is cooked fresh, or dried. The oil is use in the treatment of a number of ailments and after the oil is removed from the blubber by frying it, the processed blubber, called "crisps" is eaten as a snack.
Hazelwood told iWitness News that fishermen go out to sea every day, but only meet killer whales about four times annually.
He, however, said that when orcas are in the area, whalers could go for two months without catching pilot whales.
Barrouallie whalers have been killing short-finned pilot whales as part of a century-plus-old tradition in this multi-island nation.
In Bequia, an island in the northern Grenadines, the International Whaling Commission allows whalers to kill four humpback whales every year as part of their century-old indigenous whaling tradition, but whalers sometime go for years without a single kill.
The orcas, on Sunday, bring to nine the number of the cetaceans that Barrouallie whalers are known to have killed since 2015, when four orcas were killed.
A whaling industry insider, who asked not to be named in this article, told iWitness News on Monday that it is highly unlikely that the Gonsalves-led Unity Labour Party administration would pass a law banning the killing the orcas before the next general election.
Whaling industry workers process an orca carcass in Barouallie on Monday. (iWN photo)
Gonsalves' government is into a second consecutive term with a one-seat majority in the nation's parliament and the commenter told iWitness News that the Barrouallie whalers and their extended families are largely supporters of the ruling Unity Labour Party.
To ban the killing of orcas would amount to taking food out of the mouths of the whalers and other persons who benefit from the industry and they will punish the government at the polls for it, the industry insider told iWitness News on Monday.
Meanwhile, speaking on radio in April 2016, days after iWitness News broke the news of the killing of the orcas, Gonsalves condemned the whalers' action.
"… what he did was plain wrong. Not just because it happened in front of tourists, but [because] he must not kill the orcas."
The prime minister noted that there is an international ban on the killing of bottlenose dolphins, and killer whales and SVG has signed on to the ban.
He further noted that four killer whales were killed in the Central Leeward town in July 2015.
"And the fourth one, they had so much meat that some of the meat spoiled," Gonsalves said.
"What will happen is that legislation will be brought to stop that killing. It will be made an offence in the same way we have done with turtles," said Gonsalves, whose government banned the killing of turtles, harvesting of their eggs or disturbance of their nests, effective Jan. 1, 2017.
30 replies on "Whalers kill 3 more orcas in St. Vincent (+video)"
This is wrong in all sense. Yes you defending your lively hood and keeping your tradition alive that's understandable, but you kill three of the most beautiful creatures on earth. If you kill one it's over 2000lbs of meet that's enough to support the very few people who eat them.
It's a very bad look from all angles, you making sure you eat while the man who have his business in whale watching have to suffer. What about his lively hood is it not necessary for there kids to enjoy a comfortable life as well. It's just open the wounds wider in our tourism industry when this hits international boarders.
Q. If you gave three cattle in your back yard to help support your family, would you kill all three of them the same day knowing that the demand for the beef is not high?
Muse says:
It's really easy for us once our bellies are full to castigate other people. Those who are familiar with that area of the country knows how difficult it is to earn a living there. Unemployment is very high and the agricultural sector is a shadow of its former self and here we are criticising people for killing a mere 3 orcas for food. Am all for protecting endangered species, but until the economy improves the people are the endangered species that have to be protected first.
It's not about castigating other people it's about not being greedy and cruel. Those or orcas worth a few thousands dollars dead compare to how much they worth alive. If those orcas were the last three orcas in and around our waters someone lively hood just shutdown for something over a short term.
Why kill all three when other people have use for them also which brings me right back to my question.
. If you have three cattle in your back yard to help support your family, would you kill all three of them the same day knowing that the demand for the beef is not high
cornonthecomb says:
oh my GOD!! how anybody can defend the killing of orcas for food is amazing! ITs not only them that live in this world, WE ALL DEPEND on the OCEAN, and the OCEAN is in DANGER. Whales are a very very important part of what WE need. They are super important to the environment. "Even whale poop plays a large role in the environment by helping to offset carbon in the atmosphere.
Studies have shown that the nutrients in sperm whale poop helps stimulate the growth of phytoplankton which pull carbon from the atmosphere to provide a cleaner and healthier breathing environment for all animals.
Estimates state that as much as 400,000 tonnes of carbon are extracted from the air due to these whales each year!
In additional to feeding carbon fighting phytoplankton the fact that whale poop stimulates the growth of phytoplankton means that it also helps feed other species that feed on phytoplankton for their survival."
BESIDES whales, and especially killer whales are super similar to humans in behavior. These people are going straight to hell…and so is anybody that can defend their actions.
the government need too do something about this
Ash Brown says:
Orcas are top of the food chain predators and have high levels of heavy metals in their meat.This can't be good for human consumption.
Agustus Cart says:
The killing of these Orcas are unjustified in every sense. However, I quite understand the sentiments expressed by the fisherman. He said he asked for equipment to assist them in catching other species of fish but the fisheries department did not respond. These killing are a consequence of the indecisiveness of the Government. The legislation should have already been passed and possible alternatives identified. Unfortunately, there are likely to be repercussions these killings.
Fishing can be a very lucrative for Vincentians and can help reduce the unemployment gap. The Government needs to launch an initiative to empower more coastal residence to make their living from our valuable fisheries. Build more docks and marinas along the coast, form cooperatives, identify economies of scale for the procurement and use of fishing aparatus, and source more affordable fishing trawlers and long line fishing boats. These are simple measures that can be taken to create more opportunities for our people.
Living in the stone age. no wonder. They don't even know that rest of the world are in the second quantum revolution. What is wrong with you people? If that island was inhabited by whites it would be booming. look at Africa then look at the rest of the world. What do you see? Finally, SVG in the world news for violating and killing an endangered species. The worse part is that they don't even know what they have done. What a la la.
Mairna says:
Africa? First of all orcas are not endangered; second of all, Iceland (in Europe) is killing Fin whales, an endangered species
Why don't you say the same thing to the Japanese?
Jo Fitz says:
Disgusting Racist comment R. Faroe islands also hunt whales for their meat and guess what colour their skin is. You obviously don't understand this situation. If people have no employment and limited access to make money then they do what they do to survive. I don't agree with this but I am not in their situation. Also why is an orca better than a tuna fish or a pointer whale or for that matter a cow?? Many of us eat animals without seeing the reality of the situation. Back to my first comment….. people are not the colour of their skin. People are individuals formed through their experiences, family and environment no one is defined by their skin colour. Your racist comment just ruined my morning 🙁
Moy says:
Orcas are not endangered. Hunting orcas feed many ppl and support many families. Let's be honest, not many ppl directly benefit from whale watching tours. SVG is being bullied and unfairly targeted. SVG's whaling practices are sustainable unlike other countries like japan.
NB says:
Could you please provide info about the sustainability of SVG's whaling?
Watching you says:
9 orcas to date since 2015. Does that not tell us that they are really dependent on it as food. SVG is a tiny island. What sort of damage are they really causing on a global scale? I know nothing about SVG laws, but maybe some sort of licensing be put in place. Not sure what licensing will do but it may just deter people from whaling too much. This scenario reminds me of the situation with Dominica, where the PM indicated that no matter what environmental protection you try to have, the smaller nations will always suffer because their footprint is minuscule compare to bigger, more developed nations….and in the end they still got wiped out.
I hope Gonsalves does not ban whaling. He seems to have a 'tell it like it is' attitude. Hope he has the same strength to tell the tourists that this time, they will have to deal with the thought of people eating whale meat.
Heard it is like steak, btw….wouldn't mind trying myself…medium rare!
otmar schaedle says:
Indeed, in SVG Sea-Turtles are now protected by law since Jan 2017. However, in spite of the existing legislation the so-called 'turtle sanctuary' on Bequia, better to be named 'Turtle Prison' continues in full swing for commercial reasons as hundreds of turtle hatchlings are being poached every year to replace the hundreds of the provious haul. Most of those have since vanished! This operation should be closed down or converted into a scientifically monitored marine sanctuary, where only a few tutrles besides other marine creatures could be studied and admired.
Alyson W. (@AlysonHypnotik1) says:
The Government should ban this immediately, or a full out boycott of this island as a tourist or Cruise destination will likely come into play. These orcas are worth way more alive to the local economy drawing whale watching enthusiasts from all over the world.filling hotel rooms, restaurants, bars and beaches for years to come ,Instead these sentient, intelligent marine mammals are being hacked up on your beaches. Beaches that I am no longer interested in visiting again until a ban comes into effect,
gravitychek says:
The Canadian government and the European Union need to weigh in on this. If they honor the ban then the states that receive support from them can be influenced to enforce the ban or make whatever laws they need to make to stop this killing. Killing cetaceans is not like killing a deer. They are sentient beings not cattle. Evolve already SVG!
Vincentians don't have a mind of their own, we flip flop on emotions. We have been whaling for centuries. How many whales do we catch a year? I can see you either setting a quota or a season. This is a unique tradition to SVG, this is a form of livelihood, this is what has sustained the Barrouallie community for decades. What about the Tri Tri? These are baby fishes that are not allowed to developed to produce more Tri Tri. We can go on and on.
Then we should not kill cattle, goats, pigs, fowl, fishes…
Have you ever seen the way a cattle or a pig is killed??
Sarah Ward says:
Sentient sentient sentient. I am so tired of hearing that word. All it really means is that a creature has sensory organs. A cockroach would qualify.
uneducated monkeys killing our beautiful planet , for the sake of having another baby. Puke on these people.
Really good reporting from Kenton Chance. Maybe he should make a whole documentary film so you people can understand the reach of what you are witnessing. There are of course three sides to every story. Nonetheless, I felt disgusted to watch this. I couldn't even watch the whole video. It is not racism, it is self-criticism.
Tracey O'Donnell says:
Just stop killing !!!!!!!!!!!!!!!!!!!
Jens Jansen says:
Boycott these islands 100% and make the government there crack down on these scumbags, and stop making excuses for their filthy actions !!!
A. McKenzie says:
This fisherman is an uneducated moron in its true sense. What he knows about scientific data that the orca are not a threatened species?Scientific data is valuable to him when it counts. How about conservatiom? The blame surely falls on the government for being complacent on the who matter and is sending a wrong message. last year a tourist ship by passed St Vincent for the very reason. They were hit financially in their pockets. More of these actions need to be exercised in order to curtailed the killing of these lovely creatures.Others may say that they crime rate in Vincent is high for a small country much less to the killing of orcas. They killed human more Savagely much less to orcas.
Peter Appollo says:
We kill foul, pig ,cattle ,rabbit , goat ,,sheep,some monkeys,horse, catch fish lobster crayfish whale , shark all sort of other sort of other animals by the millions make money ,eat them as food and I have not heard seen or read any article by Kenton Chance and the animal consciences expressed their concerns for these innocent beautiful creators. HYPOCRITES
Dee Callahan says:
Stop having so many children!!!! Fewer people, fewer mouths to feed. I am so tired of people saying they can't feed their children. You should have thought about that BEFORE you had them. Feeding them mercury laden meat is a really bad idea. They grow up brain damaged, I guess that's what happened to these people. Japan, tbe Faroes, all of you, are savages. I wish nothing but harm and misfortune on all of you. Karma is a bitch, and I hope she kills you!!
All you 'sensory' people who chatting about couldn't watch video, have you ever seen a Cattle, Pig or Rabbit killed. The cattle is stabbed in the neck, the pig is sometimes stabbed or hit on the head several times, a rabbit is hit on the head. People, wake up!!!! Start thinking for your own selves. Kenton's video should not cause any stir, actually I see it as enlightening! Do we cry shame on the men who rape our women and girls? OOh, I see again it depends on who is the 'raper man!!!' Japan kills whales like how we change our clothes, has anyone of you ever cried shame????
C. ben-David says:
The relevant legislation has not been enacted simply because the electoral seat where Barrouallie is located which was won by the governing party in 2015 is still being contested in the courts.
Given that the ruling party won the 2015 national election by only one seat, if a new election is called in Barrouallie and the promised ban on orca fishing were now in place, the ruling party would likely lose the seat thereby resulting in their loss of power.
So, this is not about protecting a possibly endangered animal species; it is about protecting a potentially endangered political party.
Kenton, what was your goal in posting this video? What tourism you plp keep referring too about whale watching you honestly believe tourists come to st Vincent to watch whales… do they even come to St Vincent. Vincentians need to wake up and start thinking for themselves the white man come to our country and tell us this is not right and we go with that. This is a tradition for years in our town of barroullie embrace it! We always want to copy or have someone tell us what to do. Many countries out there have many different traditions who are you outsider to tell us this is not right. This is why many of us don't know our black history we only know what the white man tells us. Stop being slaves in your own country! Embrace who we are as a community Bagga people.
Concerned Citizen says:
23rd April 2018 at 11:12 AM
I think that many will be outraged by the videotaping of the whales being cut up on the beach. And there are many points of view on the matter. St. Vincent is a poor country and many of the people in the region of the country where this happened count on this for their livihood. If you know anything about St. Vincent, you know that tourism is not the main source of income, and folks don't go to St. Vincent to (primarily) whale watch.
I am disturbed by some of the racist and hateful comments posted by people who I'm willing to bet don't even know where St. Vincent is, and are not aware of the struggles faced by the fishermen. Before making judgement and wishing them death, take a trip to the island and speak to the people. The people here survive on fishing and agriculture. These sea creatures will provide food and income for many people for months. They are not hunting on a commercial scale. These men go out in small fishing boats, and fish every day to feed their families and communities. If you want to be outraged do a Google search on whaling and you will find that Japan, Norway and Iceland have very large whaling operations, killing thousands of whales among the three nations. Apparently Japan gives away whale meat, because they catch so many. Vincentian fishermen have killed 7 whales in 3 years (on the main land, according to this article), which are completely consumed. None of the whale goes to waste (except the intestines). This seems like a very sustainable number with little impact on the environment as a whole. This has been done for thousands of years first by the natives and now by their descendants.
Comments about the people being ignorant is uncalled for. Note that St. Vincent, as well as many other countries around he world, is a small nation. Highly educated folks usually aren't fishermen because it's very hard work. Fishing is usually a tradition passed down from generation to generation. Further, this video was shot in a rural area. I believe that if you go to rural areas of highly developed counteies you will find a similar situation. And do not equate lack of education with ignorance. I know many educated people who are quite ignorant.
It's easy to criticize when you are not in that persons shoes.
I will not comment on the remarks about politics and politicians, as this issue seems to me to be about the livelihood of a community. However, if the fishermen are not allowed to fish then there must be alternate sources of income provided for them to survive.