text:
Since its establishment in May 1966, CACCI's growth direction has been charted by committed business leaders from around the region – astute businessmen of diverse cultural and economic backgrounds but who share a common aspiration of regional economic growth through partnership between the private sector, government and international institutions.
During their respective tenure as President of the Confederation, each of these business leaders took on the important responsibility of uniting the region's business community towards a singular objective of strengthening Asian chambers, for an effective role of serving business and their respective countries, ultimately contributing to the accelerated development of the region as a whole.
Demetrio Munoz (Philippines) 1966-1968 & 1978-1979
* President of the Chamber of Commerce of the Philippines
* Transportation magnate
* President of Continental Bank
Chen-Fu Koo (Taiwan) 1968-1970
* President of the Chinese National Association of Industry & Commerce
* Chairman of Taiwan Cement
* Founding Chairman of the Straits Exchange Foundation (SEF)
* Arranged the first direct talks between Taiwan and mainland China since 1949
Too-Pyung Park (Korea) 1970-1972
* President of the Korea Chamber of Commerce & Industry
* Chairman of the Doosan Group
* Philantropist
Tekashi Rinoiye (Japan) 1973-1974
* President of the Yokohama Chamber of Commerce & Industry
Freeman Strickland (Australia) 1974-1976
* Chairman of Melbourne Chamber of Commerce & Industries
* Chairman of Norwhich Union Life & Insurance
* Spitfire Pilot during World War II
D. C. Khotari (India) 1976-1978
* President of the Federation of Indian Chambers of Commerce & Industry (FICCI)
* Chairman of the KICL Group
* Agroindustrial magnate
Fred J. Elizalde (Philippines) 1978-1979
* President of the Philippine Chamber of Commerce & Industries
* MBC media group Chairman
* "Radio King" of Philippines
Aurelio Periquet (Philippines) 1979-1980 & 1995-1996
* Member of the Monetary Board
* Prominent businessman
Jeffrey L. S. Koo (Taiwan) 1980-1982 & 1986-1988
* President of the Chinese National Association of Industry & Commerce (CNAIC)
* Chairman of the CTBC Financial Group
* Ambassador At-Large
Shigeo Nagano (Japan) 1982-1983
* President of the Japan Chamber of Commerce & Industry
* Chairman of Nippon Steel
* Trade Advisor to the Japanese Government
Noboru Gotoh (Japan) 1983-1984
* Chairman of Tokyu Railroad
* Chairman of the Pacific Basic Economic Council (PBEC)
Soo-Chang Chung (Korea) 1984-1986
Kenneth Court (Australia) 1988-1990
Business Consultant & Company Director on the board of various companies (1973-Present)
Member of the CACCI Advisory Board (1990-Present)
Trustee, The Sir Frank Ledger Charitable Trust (Present)
Chairman, All Saints' College Foundation (Present)
Chairman of the Australia Taiwan Business Council (1989-1992)
President of the Confederation of Asia-Pacific Chambers of Commerce and Industry (1988-1990)
Member of the Australian Manufacturing Council (1984-1986)
President of the Australian Chamber of Commerce (1983-1985)
President of the Perth Chamber of Commerce (1981-1983)
Foundation Chairman of the State Small Business Council
Represented Western Australia on the Federal Government Small Business Council
Member of the Western Australian Government Asia Business Council
Member of the Board, Arthritis Foundation of W. A.
Member of the Board and Chairman, All Saints' College
Executive Chairman, Asia Research Centre, Murdoch University
Directorships
Former director of a number of Australian public companies including:
Perpetual Trustees Australia Limited
Iluka Resources Ltd
Innerhadden Limited
Resource & Industry Limited
Petroleum Ventures Limited
Research Manager and Partner of DJ Carmichael & Co., sharebrokers, Perth (1966-1973)
Member of The Stock Exchange of Perth Ltd.
Research Department with Ian Patter & Co., sharebrokers, Melbourne (1964-1966)
Hale School, Perth, Western Australia
Bachelor of Economics, University of Western Australia
Associate of the Securities, Institute of Australia, A. S. I. A.
Citizen of the Year. Commerce & Industry 1989 (Western Australia) Cit. W. A.
Sub-Lieutenant, Royal Australian Naval Reserve (Retired)
Runs historic "Yarrabah" farm, Mundijong, Western Australia
Four times Australian Champion Yachtsman. Prominent in ocean yacht racing
With yachts "Prime Minister" and "Second Term".
Ir. Sotion Ardjanggi (Indonesia) 1990-1992
* President of Indonesia's KADIN
* President of Indonesia Cement
* Minister of Industry
R. P. Goenka (India) 1992-1994 & 1998-2000
* President of the Federation of Indian Chamber of Commerce & Industry (FICCI)
* Founder & Emeritus Chairman of the R.P.G. Group
* "Takeover King"
Amb. Alfonso T. Yuchengco (Philippines) 1994-1996
Honorary Chairman of the Rizal Commercial Commercial Banking Corp. (RCBC)
Ambassasor Yuchengco is one of the most successful and respected businessmen in the Philippines. Aside from banking (RCBC), the range of his family-owned businesses includes insurance, oil exploration, construction, logging, agriculture, communications and manufacturing.
His business acumen and international ties have made him a prime candidate for the diplomatic corps. During President Corazon Aquino's administration, he was appointed Philippine Ambassador to China. He also served as Philippine ambassador to Japan during President Fidel Ramos' tenure. During President Estrada's term, he was appointed presidential assistant on APEC matters. When President Macapagal-Arroyo took over in 2001, he was named roving ambassador in charge of Korea, China and Japan, and was later appointed Philippine Permanent Representative to the United Nations, a position he held until December 2002
At present, aside from his business endeavours, Amb. Yuchengco is still very much active in various civic, professional and philanthropic organizations. He is also a recipient of many distinguished awards, including The Outstanding Filipino Awardee for 2000.
Masafumi Ohnishi (Japan) 1996-1998
* Vice Chairman of the Japan Chamber of Commerce & Industry
* Chairman of the Osaka Chamber of Commerce & Industry
* Chairman of Osaka Gas
Soo-Young Lee (Korea) 2000-2004
* Chairman of OCI (Oriental Chemical Industries)
K. K. Modi (India) 2002-2004 & 2004-2006
Chairman of Modi Enterprises
President of Godfrey Phillips India Ltd.
Modi Enterprises is group of companies with international partners and engaged in various sectors including cigarettes, tea & beverages, entertainment, consumer products, network marketing, chemicals, tertiary education and Gourmet restaurants. While Godfrey Phillips India is the second largest player in the Indian domestic cigarette industry.
Mr Modi has made a significant contribution to Indian industry and holds numerous positions in the industry, trade, education, sport and charitable organizations such as President of CACCI, Former-President of FICCI; former President of PHD Chamber of Commerce and Industry, Member of the Board of Governors, Indian Institute of Management, Lucknow; and a life member of the All India Heart Foundation, Delhi.
Mr. Modi is former President of the Confederation of Asia-Pacific Chambers of Commerce and Industry, and currently carries the title of President Emeritus and member of the CACCI Advisory Board.
Harvey Chang (Taiwan) 2006-2008 & 2008-2010
Vice Chairman of Taiwan Fixed Network Co.
Mr. Chang was President and Chief Executive Officer of Taiwan Mobile Company Limited (TWM), a major telecommunication service provider in Taiwan, from 2003-2010. He served as President of the Confederation of Asia-Pacific Chambers of Commerce and Industry (CACCI) from 2006-2010.
Between 1998 and 2003, Mr. Chang was Senior Vice President and Chief Financial Officer of Taiwan Semiconductor Manufacturing Company Limited (TSMC) which has enjoyed the reputation of "Being the most admired company in Taiwan" for many years.
Mr. Chang served in Taiwan's financial service community for 20 years. Immediately before his appointment in TSMC, Mr. Chang was the Chairman of China Securities Investment Trust Corporation- the largest mutual fund management company then. Earlier than that, he was the President of China Development Corporation, one of the two development banks in Taiwan.
On an earlier assignment, he was the founding President of Grand Cathay Securities Corporation, the leading underwriter of equities and bonds in Taiwan. Before that, he was the Senior Vice President of Chiao Tung Bank, a development bank owned by the Government of the Republic of China.
Mr. Chang is recognized by Finance Asia magazine for three consecutive years, in its Annual Best Asian Companies Survey, as "the Best CFO in Taiwan" in 2001, 2002 and 2003. Other citations to accredit him include "Achievement in Best Practice Award 2001" (CFO Asia Magazine), "Best Investor Relationship Officer Asia 2001" (Investor Relations Magazine). He was awarded the Eisenhower Exchange Fellowship in 2002.
Mr. Chang is a member of the Asian Executive Board of the Wharton School and recipient of the Distinguished Service Award from Wharton School Alumni Association in 1998. In addition, he is Chairman of Felix S.Y. Chang Foundation Board, Taipei Ching Yuin Cultural & Educational Foundation Board and IC Radio Station. He has been listed in "Who's Who of Professionals" since 1997.
Mr. Chang received his MBA from the Wharton School, University of Pennsylvania in 1977; and B.S. degree from National Taiwan University in 1973.
Amb. Benedicto V. Yujuico (Philippines) 2010-2012 & 2012-2014
Chairman Emeritus, Athena e-Services Corp.
Chairman, International Trade Affairs, PCCI
Ambassador Benedicto V. Yujuico is the Immediate Past President of the Confederation of Asia Pacific Chambers of Commerce and Industry (CACCI). During his Presidency from 2010 to 2014, Amb. Yujuico met with Presidents, Prime Ministers and Senior Government officials as well as prominent businessmen from 22 CACCI member countries. He also served as Member of the General Council (Governing Body) of the ICC – World Chambers Federation, which is the ICC's specialized division for its chamber of commerce members worldwide.
Ambassador Yujuico holds a Masters Degree in Business Administration from Wharton Graduate School of the University of Pennsylvania in 1970. He has been a director of several commercial and savings banks as well as venture capital corporations on telecommunications, storage and computer devices in different countries worldwide.
His experience also includes real estate development such as IT parks, residential and commercial buildings including mixed used developments in addition to his investment in social media networks in the United States, Philippines and other countries. He is a senior level Adviser of companies in the United States, India, Brazil, and Russia.
He is the Chairman Emeritus of Athena e-Services; a technology company involved in electronic publishing, litigation support, financial research for Fortune 500 companies in the United States and other countries; and concurrently the Chairman for International Trade Affairs of the Philippine Chamber of Commerce and Industry.
Ambassador Yujuico also served as the Special Envoy of the Philippine President for Trade Relations. He has led official and private sector missions to several countries to facilitate trade and investment in the Philippines particularly in the areas of agriculture, alternative energy, business process outsourcing and real estate development.
Jemal Inaishvili (Georgia) 2014-2018
President of the Confederation of Asia Pacific Chambers of Commerce and Industry (CACCI).
He is member of the B20 International Business Advisory Council (IBAC) and member of the ICC World Chamber Federation General Council, as well as member of the Council of the Georgian Chamber of Commerce and Industry.
Prior to this, Mr. Inaishvili served as the president of the Georgian Chamber of Commerce and Industry as well as the vice president of CACCI. He dedicates most of his time to promoting Chamber movement and facilitating trade and investments in the region. Mr. Inaishvili has had extensive business experience over the years in shipping, logistics and International trade.
He is the former CEO of Poti Sea Port and former president of Black and Azov Sea Ports Association (BASPA) and at present is honorary president of this organization.For some time, Mr. Inaishvili had been actively involved in politics. He was elected member of the Georgian Parliament from his native Batumi constituency in 2005 and in the same year was elected by parliament as deputy chairman. Mr. Inaishvili is awarded with Medal of Honor of Georgia, one of the highest civilian awards of the country.
President's reports
2019 CACCI calendar
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Robo-Retro Part Four: Transformers, the Final Chapter
By: Justin Cummings
June 21, 2016 / Articles
Here we are, the last part of Robo-Retro. I began this as a way to get people excited for Voltron, but this series has become so much more to me. This has become my way to talk not just about some of my favorite shows, but the role of mechs in animation all the way up to today. Everything has been intertwining, and the franchise I've mentioned but haven't yet talked about is Transformers. Unlike something like Robotech, the vast majority of people have at least heard of Transformers, if nothing else due to the horror that Michael Bay churns out (I swear if you're going to call it Hong Kong at least edit out the Empire State Building!) The franchise however goes back much farther, and has quite the story. Like most things in Robo-Retro, it started with Japan.
Back in the 1980's, two completely separate toy lines released: microman and dioclone. Hasbro bought the rights to both separate toy lines, and then gave Jim Shooter and Dennis O'Neil the job of creating a story. What we got was Transformers, which was a TV show, comic book, and toy line beginning in 1984. It was Bob Budiansky though who came up with the specific character stories, and from the get go, the show started contradicting him. The show was quite successful though, warranting a full animated movie, where one the franchises biggest missteps was made. In order to help make way for the new waves of the toys, they killed off Optimus Prime. Nobody saw this coming, and children everywhere were left devastated. They did learn their lesson though, as Optimus was eventually brought back. The show went well, but eventually it ended.
The End of the Beginning
In the 90's however, we got a new show, Beast Wars. It was a much more animal focused show, that actually continued the story from the gen one show, but added a lot more to the universe. Several other comic books took place in this universe, but on TV gen one was over. In its place, we were getting a new era. Japan in 2000 created a series called Robots in Disguise. It was a 39 episode anime that did eventually cross over into the American shows, but it's actually been banned from United States TV. That's quite a feat for a show originally aimed at kids. It was after that though, that we got my favorite aspect of the franchise.
Known as the Unicron trilogy, Armada, Energon, and Cybertron were three 52 episode series that tied together, coming out across most of the first half of the 2000's. These shows are all held in high regard, and after these solid shows, things get sketchy. 2007 gave us Transformers Animated, which isn't as bad as it was treated by Cartoon Network. It was good, but is mostly the cult hit of the franchise. Transformers Prime came out next, and was the first full CGI series. The show introduced a main female character, which was quite impressive. It aired on the Hub, which behind the scenes was the set up for Transformers: Rescue Bots. When this show, aimed at preschoolers, came out, combined with the live action movies, the franchise seemed dead.
A Spark of Hope
Now we have hope though. The newest series, simply entitled Transformers, is a return to form, specifically the style of Transformers Animated. In addition, the High Moon Studios games are actually quite good. so although the franchise has had a tough run, it seems to be on the upswing. Thank you for sticking with me guys; I'm glad to have finally shared this with all of you. I can't wait to see what the next big Mech series is. Who knows, I may need to do a part five.
https://en.wikipedia.org/wiki/Transformers
https://en.wikipedia.org/wiki/Transformers_(film_series)
http://birthmoviesdeath.com/2013/07/29/how-optimus-primes-death-defined-a-generation
Tags: Robo-Retro, Transformers
Justin Cummings
I'm currently studying Communication at Lynchburg College with an emphasis in Electronic Media and a double minor in business and gender studies. I've been a huge fan of animation since I was little and thus have very eclectic tastes. Outside of animation I love Survivor, comic books, and a whole treasure trove of other things. More by this author »

redpajama_set_name: RedPajamaCommonCrawl


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Jen Psaki Makes Excuses for Dems Losing Florida, Insults Latino Voters in the Process
By robm November 10, 2022 Updated: November 10, 2022
The force of nature that IS Florida Governor Ron Desantis swept through the midterms like a late season hurricane, devastating leftists hopes and dreams all across the state. Desantis has handled every obstacle of his Governorship with class, intelligence, and a no-nonsense approach with only the best interest of Floridians in mind, and people noticed. Not only did Desantis win by double digit points, an unheard of margin, but he pulled literally every Republican on the ballot across the finish line with him. Florida is now officially as red as a state can get, and Floridians are better off for it.
Speaking of red, our favorite lefty redhead, Jen Psaki is back at it, making crazy statements and offending Latino voters in Florida after they made their wishes crystal clear Tuesday. Psaki offered some rather dubious, definitely insulting reasons for the red wave in Florida. Check this out.
Former White House press secretary, now MSNBC pundit, Jen Psaki is making a laughable case to discredit the Latino vote in Florida.
Psaki claims that Florida Latinos are different from all other Latinos. She claims that Florida Latinos are particularly susceptible to misinformation.
Psaki said "a massive disinformation problem in Spanish language media" fueled the GOP domination in Florida.
Those poor, stupid Florida Latinos. If only they would've watched more rich, elite white people in the media they would know better and had voted Democrat! It's such a shame white people weren't able to rescue them from their own ignorance and dreaded "misinformation". In the left's eyes, you can replace the word "Latino" with "black", "gay", or any other number of races or alleged "marginalized" people, because only rich, elite white people can save them!
How about this as a possible reason for the red wave in Florida; people love Ron Desantis. He has done an excellent job with Florida, and their people know it. Perhaps Latinos are also done with having radical policies like CRT, the sexualization of children, and the lgbtq air force rammed down their throats? It should be noted that South Florida has a heavy Cuban population, and they CERTAINLY aren't down with how many on the left sympathize with Communists like Fidel Castro. That's still a touchy subject for them.
The Latino vote is not the same everywhere! There are huge generational differences and in Florida population of older voters (more likely Rs) much larger than younger. Socialism does not play there. And there is a massive disinformation problem in Spanish language media.
— Jen Psaki (@jrpsaki) November 9, 2022
Excuses are like a certain orifice on the body that everyone has and doesn't smell very good. Psaki makes herself look like a giant one with these statements. She can make all the excuses for losing Florida and the Latino vote, but certainly she isn't helping the Dems case by essentially calling Florida Latino's stupid.
"Fox & Friends Weekend" co-host Rachel Campos-Duffy explained this situation beautifully.
"Patronizing white lady says Hispanics who vote Republican are ignorant, gullible victims of disinformation."
Maybe Florida has an outstanding Republican governor who has won the support of a record number of Latinos. Whether or not the Democrats want to admit it, that's exactly he case.
Perhaps Peppermint Psaki should have stayed with the White House. At least when she made stupid, ill-informed statements and told lies they were for her boss. Psaki was actually pretty good at that, unlike the new girl KJP. Either way, this certainly isn't going to play well in Florida, and hopefully Latino voters nationwide will realize which party actually values them and works for them. Hint; it ain't the left.


4-H at State Fair is synonymous with fun, excitement, interactive learning, and more! For 13 days, youth from around New York attend Fair as representatives of their county Cooperative Extension and 4-H offices. The 4-H Youth Building offers a dorm upstairs to accommodate youth and families attending Fair.
New York State 4-H Youth Developoment is fortunate to have continued support from the NYS Department of Agriculture & Markets and Cornell University for our strong presence at State Fair. Through our parntership, we are committed to an engaging experience for families. We highlight the best of 4-H, positive youth development, and New York itself through a focus on agriculture and animal care; public presentations; renewable energy; health and fitness; robotics; teen leadership; arts and crafts; and more.
Each couny participating in fair is assigned a 4-day rotation for exhibiting in the 4-H Youth Building. The 4-H Youth Building and our many animal barns act as showcases of youth learning and effort to make their best better. Inside the 4-H Youth Building, near Gate 4, county booths are spaces for interactive activities and exhibits for Fairgoers of all ages.
What is Positive Youth Development?
If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact onondaga@cornell.edu for assistance.


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Books / Product Reviews
K is for Killing – Daniel Easterman
by Ryan Paul · February 25, 2007
I can't for the life of me remember which the first book store I got this book from, but I'm relatively certain it was Waterstone's in Notting Hill Gate, and I'm very glad I did.
The story for K is set in 1940's America, and is based around three main characters. British agent John Ridgeforth, Vice President David Stephenson, and his wife and Laura Stephenson.
During the Depression, America has voted in a fascist government. Charles Lindbergh is the hugely popular, but ailing President. His Vice President David Stephenson is the most powerful fascist in the country, and a thoroughly dangerous and evil man. As World War Two starts, America's neutrality holds, but very thinly.
British intelligence can see that it won't be long before President Lindbergh dies, and David Stephenson becomes President, making America join in the war, siding with Nazi Germany against Great Britain. Under the highest orders and greatest secrecy, Whitehall dispatch an agent to America to assassinate David Stephenson before he comes President.
Aided in secret by Stephenson's wife Laura, John Ridgeforth becomes employed by David Stephenson as legal counsel to replace a former counsel who dies of a 'freak accident'.
Concentration camps are set up all over the United States for blacks, gypsies, Jews, atheists and a slew of other groups of society. The Klu Klux Klan are shoved into a powerful position, given approval from the government and set up in many areas of public life. America becomes a prison for the 'undesirables', muchlike 1930's Germany was for Jews and other minorities. Suspected enemies of the state are monitored by the dreaded Federal Bureau of Internal Security (FBIS) and its hardnose chief, J. Edgar Hoover. The Ku Klux Klan, whose members compose the core membership of the ruling AAA (Aryan American Alliance), rampages through the countryside, committing violence and murder with the full authority and assistance of the State.
Unbeknownst to Ridgeforth, Stephenson is already concocting a plan to assassinate Lindbergh. His co-conspirators, a concentration camp commandant named Jim Jackson, FBIS Chief Hoover, and a German diplomat named von Schillendorf, want Lindbergh eliminated before he discovers the secret alliance that has been made between German and American scientists for research on the atomic bomb. Stephenson has Lindbergh murdered, frames two Jewish prisoners and quickly is sworn in as President.
Stephenson invites Hitler to the US to witness the first test of the Nazi-Alliance atomic bomb in New Mexico. Delighted by the destructive force of the A-Bomb, Hitler is given even more to smile about when one of his US-based secret policemen provides him with proof that the President's wife is a traitor, she's been having an affair with a British agent, and Stephenson himself has been sexually abusing their daughter. Hitler confronts Stephenson with the evidence, radically shifting the balance-of-power between Nazi Germany and KKK Amerika.
This story is riveting and I absolutely loved it. This is another one that can be read over and over. Mostly whilst sitting on the toilet hahaha!
Buy 'K' by Daniel Easterman from Amazon


Opry Country Classics is performing on Thu, 26 Sep at the Ryman Auditorium, Nashville, TN. Ticket prices of country folk event vary depending on demand but right now you can get them for approximately $47. So hurry up and get your Opry Country Classics tickets before they are sold out.


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\section{Introduction}
\label{s:intro}

Shift-invariant systems play an important role in time-frequency
analysis and digital signal processing. In this
paper we consider time-discrete shift-invariant systems, by which we mean
functions $g_{m,n}$ of the form
\begin{equation}
g_{m,n}(k) = g_{m}(k - n a), \qquad k,n \in {\mathbb Z}, m = 0,1,\dots,M-1
\label{sis}
\end{equation}
where $g_{m} \in {\lsp^2(\Zst})$ and $0 < a \in {\mathbb N}$.
In the engineering literature such a system is known as {\em filter bank}.

Since the $g_{m,n}$ are in general not orthogonal, we are interested in
finding dual systems
\begin{equation}
\gamma_{m,n}(k) = \gamma_{m}(k-n a), \qquad k,n \in {\mathbb Z}, m = 0,1,\dots,M-1
\label{dualsis}
\end{equation}
such that any $f \in {\lsp^2(\Zst})$ has a ${\lsp^2(\Zst})$-convergent representation
\begin{equation}
f = \sum_{m,n} \langle f, \gamma_{m,n} \rangle g_{m,n} \,.
\label{fseries}
\end{equation}

There is a variety of research papers dealing with shift-invariant
systems in ${\Ltsp(\Rst)}$ or ${\lsp^2(\Zst})$, there is also no lack of methods claiming to
efficiently solve for the duals, those method naturally operate in
finite dimensions. But only little is known about the relation of the
finite-dimensional solutions and the original solutions of the infinite
dimensional model. For practical purposes it is not enough to ensure that an
approximation converges to the original solution, but one also should try
to provide a priori bounds on the rate of convergence.
The goal of this paper is to clarify the relation between the approximate
dual functions computed by a finite-dimensional approach and the original
dual functions in ${\lsp^2(\Zst})$.

Throughout the paper we will use two key ideas to derive our results.
We exploit the link between frame operators of shift-invariant systems
and Laurent operators with matrix valued symbol to apply the finite section
method for the proof of convergence. Furthermore we will discuss the rate
of convergence making use of results of the off-diagonal entries of the
inverse of certain types of matrices.

A detailed investigation of shift-invariant systems in ${\Ltsp(\Rst)}$ can be found
in the papers of Ron and Shen~\cite{RS95a,RS97b}. In~\cite{Jan98}
Janssen analyses duality conditions for shift-invariant systems in
${\Ltsp(\Rst)}$ and ${\lsp^2(\Zst})$ with emphasis on Gabor systems. For an overview of the
theory of filter banks the reader may consult~\cite{VK95}.

The concept of frames~\cite{DS52,You80} provides a natural environment to
study shift-invariant systems~\cite{Dau92,RS95a,Jan95}. We say that the
set $\{g_{m,n}\}$ is a frame for ${\lsp^2(\Zst})$ if there exist constants $A,B>0$
such that
\begin{equation}
\label{framedef}
A \|f\|^2 \le \sum_{m,n} |\langle f, g_{m,n} \rangle|^2 \le B \|f\|^2 \,,
\qquad \forall f \in {\lsp^2(\Zst})\,.
\end{equation}
If the shift-invariant system $\{g_{m,n}\}$ satisfies condition~\eqref{framedef}
we will also call it {\em shift-invariant frame}. The dual frame is
given by
\begin{equation}
\gamma_{m,n}(k) = (S^{-1} g_{m})(k - na)
\label{dualframe}
\end{equation}
where $S$ is the frame operator defined by
\begin{equation}
Sf = \sum_{m,n} \langle f, g_{m,n} \rangle g_{m,n} \,.
\label{frameop}
\end{equation}

Among all dual systems satisfying~\eqref{fseries} the duals defined
by~\eqref{dualframe} have minimal ${\lsp^2}$-norm. In this paper we restrict
our attention to duals with minimal ${\lsp^2}$-norm, therefore we
will be somewhat sloppy in our terminology and refer to the $\gamma_{m,n}$
defined in~\eqref{dualframe} as {\em the} duals.

We define the analysis operator $F$ by
\begin{equation}
F: f \in {\lsp^2(\Zst}) \rightarrow Ff = \{ \langle f, g_{m,n} \rangle\}_{m,n}
\label{frameanal}
\end{equation}
and the synthesis operator, which is just the adjoint operator of $F$, by
\begin{equation}
F^{\ast}: c \in {\lsp^2}({\mathbb Z} \times {\cal I}) \rightarrow
F^{\ast} c = \sum_{m,n} c_{m,n} g_{m,n}
\label{framesyn}
\end{equation}
where ${\cal I}:=\{0,1,\dots,M-1\}$. The frame operator can be written
as $S = F^{\ast} F$.

If the $g_{m}$ are constructed from a single function $g$ by modulations, i.e.,
\begin{equation}
g_{m}(k) = e^{2\pi i mk/M} g(k)
\label{gabor}
\end{equation}
we obtain the well-known {\em Gabor system} or {\em Weyl-Heisenberg
system}~\cite{Gab46,FS98}. A nice property of Gabor frames is that the dual
frame is also generated by a single function, we have~\cite{Dau90}
\begin{equation}
\gamma_{m,n}(k) =  e^{2\pi i mk/M} \gamma(k-na)\qquad \text{with}\,\,\,
\gamma = S^{-1}g\,\,.\notag
\end{equation}
In the filter bank community systems of the form~\eqref{gabor} are known
as (oversampled) DFT filter banks~\cite{BH98}.

For some special choices of $\{g_{m,n}\}$ closed-form solutions for the duals
are known~\cite{Dau90,FZ95,Jan96} (which does not necessarily mean that
those closed-form solutions can be cheaply computed in practice).
In general however the numerical computation of the dual frame elements $\gamma_{m}$
involves the solution of the system of equations~\eqref{dualframe} of
biinfinite order, independently of the choice of the basis for the matrix
representation of $S$ (be it the standard basis in ${\lsp^2(\Zst})$ or
the so-called polyphase representation used in the theory of filter banks).
We certainly cannot solve a system of biinfinite order, but we can
solve it approximately, based on a finite-dimensional model.

Whatever finite-dimensional model we choose to compute approximate duals,
we have to ask if these approximations really converge to the original solution
with increasing dimension and, whenever possible, we have to give
a priori bounds on the rate of convergence.
These are the problems we are going to attack in this paper.

Before we proceed we introduce some notation. Let $x \in {\lsp^2(\Zst})$, then
$\|x\|$ is the standard ${\lsp^2}$-norm of $x$. We define the spectrum of
an operator $T$ as usual by $\sigma(T)=\{\mu \in {\mathbb C}: \det(T-\mu I) = 0\}$.
If $T$ is positive definite and invertible then
$\|T\| = \max \{\mu: \mu \in \sigma(T)\}$ and in this case
$\mbox{cond\/}(T):=\|T\|\|T^{-1}\|$ denotes the condition number of $T$.

\section{Dual frames and the finite section method}
\label{ss:finite}

Due to the shift-invariance property~\eqref{dualsis} of the $\gamma_{m,n}$ we
only have to compute approximations to the duals $\gamma_{m} = \gamma_{m,0}$ for
$m=0,\dots,M-1$, since all other $\gamma_{m,n}$ are just translations of $\gamma_{m}$.
Equations~\eqref{dualframe}
and~\eqref{frameop} tell us that all shifted copies $g_{m,n}$
are required for the computation of the duals $\gamma_{m}$. But if the $g_{m}$ have
nice decay properties, such that most of their energy is contained in
some interval around the origin, say, then one may hope that the $\gamma_{m}$
will have the same properties. One could further argue that using only those
functions $g_{m,n}$ ``living'' in some interval around the origin, $[-N,N]$ say,
should be sufficient to compute a good approximation to $\gamma_{m}$.

Let us put these vague ideas in a more precise mathematical setting.
For $N \in {\mathbb N}$ define the orthogonal projections $P_N$ by
\begin{equation}
\label{defP}
P_N x = (\dots, 0,0, x_{-N}, x_{-N+1},\dots , x_{N-1}, x_N, 0,0, \dots)
\end{equation}
and identify the image of $P_N$ with the $2N+1$-dimensional space
${\mathbb C}^{N+1}$. We construct a finite-dimensional approximation $S_N$ of the
frame operator $S$ via the truncated frame elements $P_N g_{m,n}$ by
\begin{equation}
\label{pnspn}
S_N = (P_N F^{\ast})(F P_N)
\end{equation}
where $F$ is defined in~\eqref{frameanal}.
The $N$-th approximation $\gamma_{m}^{(N)}$ to $\gamma_{m}$ is then given by the
solution of the finite-dimensional system of equations $S_N \gamma_{m}^{(N)} = P_N g_{m}$.
In other words we truncate {\em all} frame elements $g_{m,n}$ to
the interval $[-N,N]$.

\noindent
The questions that arise with such an approximation scheme are:\\
(i) Does $\gamma_{m}^{(N)}$ converge to $\gamma_{m}$ for $N \rightarrow \infty$? \\
(ii) If the $g_{m}$ satisfy certain decay conditions, can we give an estimate
on the rate of convergence? In other words: given $\delta>0$, can we provide
an a priori estimate for the size of the interval $[-N,N]$, such that
$\|\gamma_{m} - \gamma_{m}^{(N)}\| \le \delta$, where $N$ depends on $\delta$ and the
decay of $g_{m}$?

In numerical procedures people often use a periodic finite-dimensional
model in order to approximate $\gamma_{m}$. Instead of the truncated
functions $P_N g_{m}$ (which could be thought of as padded with
zeros beyond the interval $[-N,N]$), the $P_N g_{m}$ are extended
periodically across the interval $[-N,N]$. In other words all
computations are done in the ring ${\mathbb Z}_{2N+1} = {\mathbb Z}\!\! \mod \! (2N+1)$.
In Section~\ref{s:periodic} we will investigate the behavior of the
periodic approximate dual functions, when the length of the
period approaches infinity.

\subsection{The finite section method and Laurent operators}
\label{ss:laurent}

The {\em finite section method} or {\em projection method} is a classical
technique to approximate the solution of infinite Toeplitz-type systems.
For an introduction and many variations of the theme the reader is
referred to the books of Gohberg and Fel'dman~\cite{GF74}, B\"ottcher and
Silbermann~\cite{BS90} and Hagen, Roch, and Silbermann~\cite{HRS95}.
Since we use the finite section method as a main tool
throughout the paper, we briefly describe it in the specific context
we will apply it.

Let ${\mathbb T}$ be the complex unit circle and let $\tau(z)$ be a continuous
function on ${\mathbb T}$, $\tau \in {\boldsymbol C}({\mathbb T})$. Let $\tau_{k} \in {\mathbb C}$ stand for the
$k$-th Fourier coefficient of $\tau$,
$$
\tau_{k} = \int \limits_{0}^{1} \tau(e^{2\pi i \omega})
e^{-2\pi ik \omega}\,d\omega
$$
then the operator $T(\tau)$ which acts on functions $x \in {\lsp^2(\Zst})$ by
\begin{equation}
(T x)_k = \sum_{j \in {\mathbb Z}} \tau_{k-j} x_j \,,\qquad  k=0,1,\dots \,,
\label{laurentop}
\end{equation}
is called {\em Laurent operator} with {\em generating function} or
{\em symbol} $\tau$. A Laurent operator may also be characterized as a
bounded linear operator acting on ${\lsp^2(\Zst})$ which commutes with the forward
shift operator on ${\lsp^2(\Zst})$, cf.~Chapter~13.2 in~\cite{GGK93}.
Since a Laurent operator can be represented
by a biinfinite Toeplitz matrix, we will often identify $T(\tau)$
with its matrix representation.

Consider the operator equation $T x =y, x,y \in {\lsp^2(\Zst})$ or, in matrix
representation
\begin{gather}
\label{toepsys}
\begin{bmatrix}
\ddots & \vdots   & \vdots   & \vdots  &         \\
\dots  & \tau_{0}  & \tau_{1}  & \tau_{2} & \dots   \\
\dots  & \tau_{-1} & \tau_{0}  & \tau_{1} & \dots   \\
\dots  & \tau_{-2} & \tau_{-1} & \tau_{0} & \dots   \\
       & \vdots   & \vdots   & \vdots  & \ddots  \\
\end{bmatrix}\,\,
\begin{bmatrix}
\vdots \\ x_{-1} \\ x_0 \\ x_1 \\ \vdots
\end{bmatrix}
\begin{bmatrix}
\vdots \\ y_{-1} \\ y_0 \\ y_1 \\ \vdots
\end{bmatrix}
\end{gather}

For the approximate solution
of the biinfinite Toeplitz system in~\eqref{toepsys} by the finite section
method one considers the finite linear equations
\begin{equation}
T_{N} x^{(N)} = y^{(N)}
\label{fineq}
\end{equation}
where $y^{(N)}=(y^{(N)}_{-N},\dots,y^{(N)}_{N})$ and where $T_{N}$ are the
finite sections of the biinfinite Toeplitz matrix in~\eqref{toepsys}:
\begin{gather}
T_{0} = (\tau_{0})\,, \,
T_{1} =
\begin{bmatrix}
\tau_{0}  & \tau_{2}  & \tau_{2} \\
\tau_{-1} & \tau_{1}  & \tau_{1} \\
\tau_{-2} & \tau_{-1} & \tau_{0} \\
\end{bmatrix}
\,, \, T_{2} =
\begin{bmatrix}
\tau_{0}  & \tau_{1}  & \tau_{2}  & \tau_{3}  & \tau_{4} \\
\tau_{-1} & \tau_{0}  & \tau_{1}  & \tau_{2}  & \tau_{3} \\
\tau_{-2} & \tau_{-1} & \tau_{0}  & \tau_{1}  & \tau_{2} \\
\tau_{-3} & \tau_{-2} & \tau_{-1} & \tau_{0}  & \tau_{1} \\
\tau_{-4} & \tau_{-3} & \tau_{-2} & \tau_{-1} & \tau_{0}
\end{bmatrix}
\, \dots \notag
\end{gather}

Using the the orthogonal projections defined in~\eqref{defP}
and identifying the image of $P_N$ with the $2N+1$-dimensional space
${\mathbb C}^{N+1}$ we can express the equations~\eqref{toepsys} in the form
\begin{equation}
P_N T P_N x^{(N)} = P_N y\,.
\label{}
\end{equation}
We say that the finite section method is {\em applicable} to $T$, if
beginning with some $N \in {\mathbb N}$ for each $y \in {\lsp^2(\Zst})$ the equation
\begin{equation}
P_N T P_N x^{(N)} = P_N y
\label{applicable}
\end{equation}
has a unique solution $x^{(N)} \in \mbox{Im}\,P_N$ and as
$N \rightarrow \infty$ the vectors $x^{(N)}$ tend to the solution of
$T x=y$.

Now let $\tau$ be a continuous $a \times a$-matrix valued function
on ${\mathbb T}$, $\tau \in {\boldsymbol C}_{a \times a}({\mathbb T})$. Let
$\tau_{k} \in {\mathbb C}^{a \times a}$ stand for the $k$-th Fourier
coefficient of $\tau$, then the operator $T$ defined by the action
\begin{equation}
(T x)_k = \sum_{j \in {\mathbb Z}} \tau_{k-j} x_j \,, \qquad x \in {\lsp^2}_{a}
\label{blocklaurent}
\end{equation}
is called a {\em block Laurent operator} with generating function $\tau$.
A block Laurent operator can be represented by a biinfinite block Toeplitz
matrix, where the $\tau_{k}$ in~\eqref{toepsys} are $a \times a$
matrices. Analogous to above, for the approximate solution of biinfinite
block Toeplitz systems we consider finite sections of the form
\begin{equation}
P_N T P_N x^{(N)} = P_N y\,.
\label{}
\end{equation}

It is a well known fact and a direct consequence of the definition of
shift-invariant systems, that the associated frame operator $S$ is a
block Laurent operator, since $S$ commutes with the shift operator
$U_{a} (\phi_k)_{k \in  {\mathbb Z}} := (\phi_{k-a})_{k \in {\mathbb Z}}$.
Hence the operator equation $S\gamma_{m} =g_{m}$ can be expressed as a
biinfinite block Toeplitz system of equations,
where the blocks of the system matrix are of size $a \times a$.
The blocks themselves are not Toeplitz. It is exactly this
block-Laurent structure of $S$ that is utilized in the
polyphase representation proposed in the filter bank theory~\cite{VK95}.

Let $S_N$ be the frame operator associated with the truncated
frame elements $P_N g_{m,n}$ as defined in~\eqref{pnspn}. Since
$S_N = P_N F^{\ast} F P_N = P_N S P_N$, the corresponding
system of equations $S_N \gamma_{m}^{(N)}= g_{m}^{(N)}$ is just a finite section of the
biinfinite system $S\gamma = g$
Hence it is natural to ask if we can compute approximate the duals $\gamma_{m}$
by solving
\begin{equation}
P_N S P_N \gamma_{m}^{(N)} = P_N g
\label{framefin}
\end{equation}
for increasing $N$, or in other words, we want to know if the finite
section method is applicable to $S$. Moreover we are
also interested in the rate of convergence, in case the finite section
method applies to $S$.

\if 0
In applications the $g_{m}$ are often designed to be compactly
supported (FIR filter banks). Note that compact support
of the $g_{m}$ does not imply compact support of the duals.
For Gabor frames B\"olcskei has shown~\cite{Boe97} that
the minimal norm duals are only compactly supported in very specific
cases. It is well known that the redundancy of frames provides some
design freedom for the dual systems, which for instance can be used
to construct dual functions with compact support~\cite{WR90,DLL95}.
We come back to this topic in Section~\ref{otherdual}.
Compact support of the $g_{m}$ allows us to compute the transform
coefficients $\langle f, g_{m,n} \rangle$ exact and also the matrix
entries of $S$ can be calculated by finite sums. If the $g_{m,n}$
are not compactly supported, they usually have fast decay (which means
that they can be approximated by compactly supported functions).

Hence if not otherwise mentioned we assume that the $g_{m}$ have compact support
such that $g(k) = 0$ for $|k|>s$. Then we can compute the entries
of $S$ exactly and moreover $S$ is a $2s$-banded matrix, i.e.,
$S_{k,l} = 0$ for $|k-l|>2s$.
But we still have to solve an infinite dimensional system of equations.
Compact support of the functions $g_{m,n}$ does in general not imply compact
support of the duals $\gamma_{m,n}$, but we will see that the $\gamma_{m,n}$ will decay fast
in this case. This fact will enable us to compute finite-dimensional
approximate duals which will converge fast to $\gamma_{m,n}$ wth increasing
dimension.

The {\em projection method} or {\em finite section method} is
a classical technique to approximate the solution of
infinite (block-)Toeplitz systems. For an introduction and
many variations of the theme the reader is referred to the work
of Gohberg and coworkers (e.g., see~\cite{GF74}) and the
book of B\"ottcher and Silbermann~\cite{BS90}.

Assume that ${\boldsymbol X}$ and ${\boldsymbol Y}$ are complex Banach spaces,
$\{P_{\tau}\}_{\tau \in \Omega}$ is a family of projections acting on
${\boldsymbol X}$ and $\{Q_{\tau}\}_{\tau \in \Omega}$ is a family of projections
acting on ${\boldsymbol Y}$, where $\Omega$ is a subset of ${\mathbb R}$ and
$\sup \Omega = +\infty$. We assume that
\begin{equation}
P_{\tau}x \rightarrow x \,\,\,(x \in {\boldsymbol X}), \quad
Q_{\tau}y \rightarrow y \,\,\,(y \in {\boldsymbol Y}) \,\,\,\text{for}\,\, \tau \rightarrow
\infty\,.
\label{projections}
\end{equation}
We denote by ${\boldsymbol{\mathcal L}}({\boldsymbol X},{\boldsymbol Y})$ the space of bounded linear operators
acting from ${\boldsymbol X}$ into ${\boldsymbol Y}$. Let $A \in {\boldsymbol{\mathcal L}}({\boldsymbol X},{\boldsymbol Y})$ be given.
We say that the projection method relative to $\{P_{\tau},Q_{\tau}\}$
is {\em applicable} to $A$, if beginning with some $\tau_0 \in \Omega$
for each $y \in {\boldsymbol Y}$ the equation
 \begin{equation}
Q_{\tau} A P_{\tau} x = Q_{\tau} y \qquad (\tau \ge \tau_0)
\label{applicable}
\end{equation}
has a unique solution $x \in \mbox{Im}\,P_{\tau}$ and as
$\tau \rightarrow \infty$ the vectors $(\dots,0,x^{(N)}_{-N},\dots,x^{(N)}_N$
tend to a solution of $Ax=y$. In this case we write
$A \in \Pi\{P_{\tau},Q_{\tau}\}_{\tau \in \Omega}$.

Given a linear space ${\boldsymbol H}$ and denote by ${\boldsymbol H}_{N}$ the linear space
of column-vectors of length $N$ with components from ${\boldsymbol H}$ and let
${\boldsymbol H}_{N \times N}$ stand for the space of $N \times N$ matrices with entries from
${\boldsymbol H}$.

The following result is due to Gohberg and Kaashoek, it is
stated as a remark in ~\cite{GK94} and is a consequence
of Theorem~4.1 in~\cite{GK94}.

\begin{proposition}
Let ${\boldsymbol H}$ be a separable Hilbert space, and let $A$
be an invertible block Laurent operator on ${\lsp^2}({\boldsymbol H})$, defined by
$A=[\psi_{i-j}]_{i,j=-\infty}^{\infty}$, where $\psi$ is a
continuous ${\boldsymbol{\mathcal L}}({\boldsymbol H})$-valued function on the unit circle.
Let $P_N (N\ge 1)$ be as in~\eqref{defP}. Then the projection method
relative to $(P_N,P_N)$ is applicable to $A$ if and only if the
block Toeplitz operators $T = [\psi_{i-j}]_{i,j=0}^{\infty}$ and its associate
$\tilde{T} =[\psi_{j-i}]_{i,j=0}^{\infty}$ are invertible.
\end{proposition}
\fi

\section{Rate of convergence using the finite section method}
\label{s:rate}

If the functions $g_{m}$ have a certain rate of decay, one would expect
that the dual functions also share this behavior.
We consider the following three types of decay for $g_{m}$ and study
first how a certain decay property translates to decay properties of
the frame operator $S$.
\begin{itemize}
\item[a)] {\em Exponential decay}: $|g_{m}(k)| \le c \lambda^{\alpha |k|}$
      for $0 < \alpha <1$, $\lambda \in (0,1)$ and some constant
      $c$.
\item[b)] {\em Polynomial decay}: $|g_{m}(k)| \le c (1+|k|)^{-\alpha}$ for
      $\alpha >1$ and a constant $c$.
\item[c)] {\em Compact support}: $g_{m}(k) = 0$ for $|k| > s$.
\end{itemize}

The following proposition is a simple modification of well-known
results used in connection with the construction of splines and wavelet
bases.
\begin{proposition}
\label{proppolexpdecay}
Assume that the functions $g_{m,n}, m=0,\dots,M-1$ constitute a
shift-invariant frame for ${\lsp^2(\Zst})$ with frame operator $S$ and duals $\gamma_{m}^{(N)}$.\\
(a) Exponential decay: If $|g_{m}(k)| \le c_1 \lambda^{\alpha_1 |k|}$
    for $0 < \alpha_1 < \alpha <1$ and $\lambda \in (0,1)$. Then there
    exists an $\alpha_2 < \alpha$ and a constant $c_2(\alpha_2)$ such that
    \begin{equation}
    \label{exponentialmatrix}
    |S_{k,l}| \le c_2(\alpha_2) \lambda^{\alpha_2 |k-l|} \qquad
    \mbox{for} \,\,0 < \alpha_2 \le \alpha\,.
    \end{equation}
    Further it holds for the duals $\gamma_{m}$
    \begin{equation}
    \label{dualexp}
    |\gamma_{m}(k)| \le c_3(\alpha_3) \lambda^{\alpha_3 |k|}\,.
    \end{equation}
    for some $\alpha_3< \alpha$ and a constant $c_3(\alpha_3)$.\\
(b) Polynomial decay: If $|g_{m}(k)| \le c (1+|k|)^{-\alpha}$ for
    $\alpha >1$, then there exists a constant $c_1$ such that
    \begin{equation}
    \label{polynomialmatrix}
    |S_{k,l}| \le c_1 (1+|k-l|)^{-\alpha}\,.
    \end{equation}
    Further there is a constant $c_2$ such that
    \begin{equation}
    \label{dualpol}
    |\gamma_{m}(k)| \le c_2 (1+|k|)^{-\alpha}\,.
    \end{equation}
\end{proposition}
\begin{proof}
The exponential decay~\eqref{exponentialmatrix} and the polynomial
decay~\eqref{polynomialmatrix} respectively, follow from Proposition~1
in~\cite{Jaf90} by Jaffard. Proposition~2  and Proposition~3 in~\cite{Jaf90}
imply formulas~\eqref{dualexp} and~\eqref{dualpol}.
\end{proof}

We admit that~\eqref{dualexp} and~\eqref{dualpol} are not
very satisfactory from a practical point of view, since
both estimates do not provide explicit expressions for the involved constants.
We hope to address this problem in our future work.

Fortunately for compactly supported $g_{m}$ we can do better.
Compact support of the $g_{m}$ allows us to compute the transform
coefficients $\langle f, g_{m,n} \rangle$ exactly and also the matrix
entries of $S$ can be calculated by finite sums. In filter bank design
the analysis filters are often designed to be compactly supported
(FIR filter banks). Note however that compact support
of the $g_{m}$ does not imply compact support of the duals.
For Gabor frames B\"olcskei has shown~\cite{Boe97} that
the minimal norm duals are only compactly supported in very specific
cases. It is well known that the redundancy of frames provides some
design freedom for the dual systems, which for instance can be used
to construct dual functions with compact support~\cite{WR90,DLL95,Boe97}.
The following results allow us to obtain a good estimate for the
decay of the duals $\gamma_{m,n}$ for compactly supported $g_{m,n}$.
But before we can state the results we need some preparation.

Recall the a matrix $T$ is called $m$-banded, if $T_{k,l} = 0$ for $|k-l|>s$.
The following theorem about the decay of the inverse of a band matrix
is due to Demko, Moss, and Smith~\cite{DMS84}.
\begin{theorem}
\label{thdemko}
Let $T$ be a positive definite, $m$-banded, bounded and boundedly
invertible matrix in ${\lsp^2}(I)$, where ${\cal I}={\mathbb Z}, {\mathbb Z}^+$ or
$\{0,1,\dots,N-1\}$. Let $B = \|T\|$ and
$A = \|T^{-1}\|$ and set $\kappa = B/A, q = \frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}$
and $\lambda = q^{\frac{1}{m}}$. Then we have
\begin{equation}
|T^{-1}_{k,l}| \le D \lambda^{|k-l|}
\label{banddecay}
\end{equation}
where
\begin{equation}
D = \frac{1}{A} \max\{1,\frac{(1+\sqrt{\kappa})^2}{2\kappa} \}
\label{lambdadef}
\end{equation}
\end{theorem}
Further we shall need following corollary
\begin{corollary}
\label{cordecay}
Let $T$ be a matrix whose entries decay exponentially off the diagonal,
i.e.,
\begin{equation}
|T_{k,l}| \le c \lambda^{|k-l|} \qquad k,l \in {\mathbb Z}
\label{matexpdecay}
\end{equation}
and let $y$ be a sequence in ${\lsp^2(\Zst})$ with $\|y\|=1$ and with compact support
such that $y_k= 0$ for $|k| >s$. Denote $\phi = Ty$, then it holds
\begin{equation}
|\phi_k| \le \frac{c \lambda^{-s}}{1-\lambda} \lambda^{|k|} \,.
\label{vecxpcdecay}
\end{equation}
\end{corollary}
\begin{proof}
We have
\begin{equation}
\phi_k = \sum_{l=-\infty}^{\infty} T_{k,l} y_{l}
 = \sum_{l=-s}^{s} T_{k,l} y_{l}  \qquad k \in {\mathbb Z}  \notag
\end{equation}
For reasons of symmetry we can restrict ourselves to the case
$\phi_k$ with $k \ge 0$. First we consider the case $k\ge s$:
\begin{equation}
\label{case1}
|\phi_k| \le \sum_{l=-s}^{s} c \lambda^{|k-l|} = c \lambda^k \sum_{j=0}^{2s} \lambda^{s-j}
 = c \lambda^{k} \frac{\lambda^{-s} - \lambda^{s+1}}{1-\lambda} \notag
\end{equation}
Now we consider the case $0 \le k < s$:
\begin{gather}
|\phi_k| \le \sum_{l=-s}^{s} c \lambda^{|k-l|} = \sum_{l=-s}^{k-1} c \lambda^{k-l}
+\sum_{l=k}^{s} c \lambda^{l-k}  \notag\\
= c \lambda^{k+s} \sum_{j=0}^{k+s-1} \lambda^{-j} + c \sum_{j=0}^{s-k}\lambda^{j}
\notag\\
= c \lambda^{k+s} \frac{1-\lambda^{k+1}}{\lambda^{k+s+1}-\lambda^{k+s}} +
  c\frac{1-\lambda^{s-k+1}}{1-\lambda}  \notag\\
= c \lambda^{k} \frac{\lambda^{-k+1} - \lambda^{s+1}}{1-\lambda} +
  c \lambda^{k} \frac{\lambda^{-k} - \lambda^{s-2k+1}}{1-\lambda}  \notag\\
= c \lambda^{k} \frac{\lambda^{-k+1}-\lambda^{s+1}+\lambda^{k}-\lambda^{s-2k+1}}{1-\lambda}
\notag
\end{gather}
Now we show that for $0 \le k \le s$ it holds
\begin{equation}
c \lambda^{k} \frac{\lambda^{-k+1}-\lambda^{s+1}+\lambda^{k}-\lambda^{s-2k+1}}{1-\lambda}
\le c \lambda^{k} \frac{\lambda^{-s} - \lambda^{s+1}}{1-\lambda} \,. \notag
\label{SNIydecay}
\end{equation}
Set $h(x) = \lambda^{-x} + \lambda^{x+1} -\lambda-1$ for $x=0,1,\dots$ and note that
\begin{align}
                        \lambda^{-(x+1)} &> \lambda^{x+1}  \notag\\
           \Leftrightarrow \lambda^{-(x+1)}(1-\lambda) &> \lambda^{x+1}(1-\lambda) \notag\\
       \Leftrightarrow \lambda^{-(x+1)} - \lambda^{-x} &> \lambda^{x+1} -\lambda^{x+2}\notag \\
\Leftrightarrow \lambda^{x+2}+\lambda^{-(x+1)}-\lambda -1 &> \lambda^{x+1}+\lambda^{-x}-\lambda -1 \notag
\end{align}
whence $h(x)$ is strictly monotonically increasing. Since $h(0)=0$ and
by setting $x= s-k$ it readily follows that
$$\lambda^{s+1}-\lambda^{-k+1}+\lambda^{s-2k+1}-\lambda^{-k} \le \lambda^{s+1}- \lambda^{-s}$$
which implies \eqref{SNIydecay} and hence
\begin{equation}
|\phi_k| \le c \lambda^{|k|} \frac{\lambda^{-s}}{1-\lambda} \qquad \quad k \in {\mathbb Z}\,.
\label{matvecdecay}
\end{equation}
\end{proof}

Now it is easy to estimate the decay of the dual functions $\gamma_{m,n}$ for
compactly supported $g_{m}$.
\begin{proposition}
Let $\{g_{m}\}$ be a shift-invariant frame for ${\lsp^2(\Zst})$ with $\|g_{m}\|=1$
and assume that $g(k) = 0$ for $|k|>s$. Let $A,B$ be the frame bounds and set
$\kappa = B/A$ and $q = \frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}$
Then it holds
\begin{equation}
\label{dualband}
|\gamma_{m}(k)| \le C \lambda^{|k|} \frac{\lambda^{-s}}{1-\lambda}\,.
\end{equation}
where $\lambda = q^{\frac{1}{2s}}$ and
$C = \frac{1}{A} \max\{1,\frac{(1+\sqrt{\kappa})^2}{2\kappa}\}$.
\end{proposition}
\begin{proof}
Observe that $S$ is a $2s$-banded matrix. Now the result is an immediate
consequence of Theorem~\ref{thdemko} and Corollary~\ref{cordecay}.
\end{proof}

We are now able to answer the question about the approximation of dual
shift-invariant frames by duals computed via the finite section method.
\begin{theorem}
\label{theorem1}
Assume that $\{g_{m,n}\}$ is a shift-invariant frame for ${\lsp^2(\Zst})$ with
$\|g_{m}\|=1$ and frame bounds $A,B$. Set $S_N = P_N S P_N$ and
$\gamma_{m}^{(N)} = S^{-1}_N P_N g_{m}$ for $m=0,1,\dots,M-1$ and denote $\kappa=B/A$.
Then it holds:\\
(i) The finite section method is applicable to $S$ and
  $\gamma_{m}^{(N)}$ converges to $\gamma_{m}$ for $N \rightarrow \infty$. \\
(ii) If the $g_{m}$ are compactly supported with $g_{m}(k) = 0$
  for $|k|>s$ and $N>2s$, then the rate of convergence can be estimated by
  \begin{equation}
  \|\gamma_{m} -\gamma_{m}^{(N)} \| \le  \sqrt{2} C \lambda^N (\lambda^{s}-\lambda^{s+1})^{-3}
  \label{gogondiff}
  \end{equation}
  where $q = \frac{\sqrt{\kappa} -1}{\sqrt{\kappa}+1}$, $\lambda = q^{\frac{1}{2s}}$ and
  \begin{equation}
  \label{constantdef}
  C = \frac{1}{A} \max\{2\kappa,(1+\sqrt{\kappa})^2 \}\,.
  \end{equation}
(iii) If $|g_{m}(k)| \le c_1 \lambda^{\alpha_1 |k|}$
  for $0 < \alpha_1 < \alpha <1$ and $\lambda \in (0,1)$, then there exists
  an $\alpha_2 < \alpha$ and a constant $c_2(\alpha_2)$ such that
  \begin{equation}
  \|\gamma_{m} -\gamma_{m}^{(N)} \| \le \kappa c_2(\alpha_2) \lambda^{\alpha_2 N}\,.
  \label{gogondiffexp}
  \end{equation}
(iv) If $|g_{m}(k)| \le c (1+|k|)^{-\alpha}$ for $\alpha >1$, then
  there exists a constant $c_1$ such that
  \begin{equation}
  \|\gamma_{m} -\gamma_{m}^{(N)} \| \le \kappa c_1 (1+|N|)^{-\alpha}\,.
  \label{gogondiffpol}
  \end{equation}
\end{theorem}

\begin{proof}
(i) Recall that $S$ is a block Laurent operator.
It follows from Theorem~9.3 in~\cite{GF74} by Gohberg and Fel'dman
or alternatively from Theorem~4.1 in~\cite{GK94}
by Gohberg and Kaashoek that $S^{-1}_N$ converges to $S^{-1}$ strongly, i.e.,
$$\|S_N x - S x\| \rightarrow 0 \quad \forall x \in {\lsp^2(\Zst}), \,\,
\text{for} \,\,N \rightarrow \infty$$ if the generating function of $S$
is continuous on the unit circle ${\mathbb T}$ and
if the operators $Q S Q$ and $Q S^* Q$ are invertible on ${\lsp^2}({\mathbb Z}_{+})$,
where the projection $Q$ is defined by
$$Qx = (\dots,0,x_0,x_1,\dots)\,.$$
The invertibility of $QSQ$ is a direct consequence of the
positive-definiteness of $S$, since for $x\in \mbox{Im}\,Q, \|x\|=1$,
we have $\langle QSQ x,x \rangle = \langle S x, x \rangle > 0$,
hence $QSQ$ is invertible. In fact using the same argument we conclude
that $S_N$ is invertible for all $N=0,1,2,\dots$.

The continuity of the generating function of $S$ follows from the fact
that $S$ is banded, thus the generating function is a trigonometric matrix
polynomial and therefore continuous on the unit circle ${\mathbb T}$. Hence the
projection method is applicable to $S$ for any right hand side $\in {\lsp^2(\Zst})$
and the first part of the theorem is proved.

In order to estimate the rate of convergence for (ii), (iii) and (iv)
consider
\begin{align}
\|\gamma_{m} - \gamma_{m}^{(N)}\| & = \|S^{-1} g_{m} - S^{-1}_N P_N g_{m}\| =
\|S^{-1} g_{m} - S^{-1} S S^{-1}_N P_N g_{m}\| \notag \\
& \le \|S^{-1}\| \left(\|g_{m} - S_N S^{-1}_N P_N g_{m}\| +
   \|S_N S^{-1}_N P_N g_{m} - S S^{-1}_N P_N g_{m}\| \right) \notag \\
& \le \|S^{-1}\| \left( \|g_{m} - P_N g_{m}\| +
  \|(S_N - S) S^{-1}_N P_N g_{m} \| \right)\notag \\
& \le A^{-1} \|(S_N - S) S^{-1}_N P_N g_{m} \|
\label{mainineq}
\end{align}

(ii) Recall that $S$ is a $2s$-banded matrix, i.e.,
$S_{k,l} = 0$ for $|k-l| > 2s$. Applying Theorem~\ref{thdemko} we can
estimate the decay of the entries of $S^{-1}$ by
\begin{equation}
\label{frameinvdecay}
|S^{-1}_{k,l}| \le D \lambda^{|k-l|}
\end{equation}
where $D$ and $\lambda$ are as in Theorem~\ref{thdemko} with $m=2s$.
Let $\mu^{(N)}_0,\dots,\mu^{(N)}_{N-1}$ be the eigenvalues of $S_N$.
Since $S_N$ is Hermitian positive definite, it follows from Cauchy's
Interlace Theorem~\cite{HJ94} that the eigenvalues
$\mu^{(N-1)}_0,\dots,\mu^{(N-1)}_{N-1}$ of $S_{N-1}$ satisfy
$$\mu^{(N)}_0 \le \mu^{(N-1)}_0 \le \mu^{(N)}_1 \le \dots \le \mu^{(N-1)}_{N-2}
\le \mu^{(N)}_{N-1}\,,$$
which implies
$$ \mbox{cond\/}(S_N) \le \mbox{cond\/}(S)\,.$$
Hence we can use the same $\lambda$ and the same constant $D$ in order to
bound the decay of the entries of $S^{-1}_N$ independently of $N$ and obtain
\begin{equation}
\label{frameinvdecay1}
|(S^{-1}_N)_{k,l}| \le D \lambda^{|k-l|} \,.
\end{equation}

Set $\phi^{(N)}_m = S^{-1}_N P_N g_{m}$ then it follows from Lemma~\ref{cordecay} that
\begin{equation}
|(\phi^{(N)}_m)_{k}| \le D \lambda^{|k|} \frac{\lambda^{-s}}{1-\lambda} \,\,,\qquad \quad
k \in {\mathbb Z}\,.
\label{matvecdecay1}
\end{equation}
\if 0
Now we estimate the decay of the entries of $S^{-1}_N P_N g$.
Set $\phi = S^{-1}_N P_N g$ then
\begin{equation}
\phi_k = \sum_{l=-\infty}^{\infty}(S^{-1}_N)_{k,l} (P_N g)_{l}
 = \sum_{l=-s}^{s}(S^{-1}_N)_{k,l} (P_N g)_{l} \,\,, \qquad k \in {\mathbb Z}
\label{}
\end{equation}
For reasons of symmetry we can restrict ourselves to the case $k \ge 0$.
Further observe that it is only necessary to consider the case $k \ge s$
(or actually $k \ge N$?)
First we consider the case $k\ge s$:
\begin{equation}
\label{case1}
|\phi_k| \le \sum_{l=-s}^{s} D \lambda^{|k-l|} = D \lambda^k \sum_{j=0}^{2s} \lambda^{s-j}
 = D \lambda^{k} \frac{\lambda^{-s} - \lambda^{s+1}}{1-\lambda}
\end{equation}
Now we consider the case $0 \le k < s$:
\begin{gather}
|\phi_k| \le \sum_{l=-s}^{s} D \lambda^{|k-l|} = \sum_{l=-s}^{k-1} D \lambda^{k-l}
+\sum_{l=k}^{s} D \lambda^{l-k} \\
= D \lambda^{k+s} \sum_{j=0}^{k+s-1} \lambda^{-j} + D \sum_{j=0}^{s-k}\lambda^{j} \\
= D \lambda^{k+s} \frac{1-\lambda^{k+1}}{\lambda^{k+s+1}-\lambda^{k+s}} +
  D\frac{1-\lambda^{s-k+1}}{1-\lambda} \\
= D \lambda^{k} \frac{\lambda^{-k+1} - \lambda^{s+1}}{1-\lambda} +
  D \lambda^{k} \frac{\lambda^{-k} - \lambda^{s-2k+1}}{1-\lambda} \\
= D \lambda^{k} \frac{\lambda^{-k+1}-\lambda^{s+1}+\lambda^{k}-\lambda^{s-2k+1}}{1-\lambda}
\label{}
\end{gather}
Now we show that for $0 \le k \le s$ it holds
\begin{equation}
D \lambda^{k} \frac{\lambda^{-k+1}-\lambda^{s+1}+\lambda^{k}-\lambda^{s-2k+1}}{1-\lambda}
\le D \lambda^{k} \frac{\lambda^{-s} - \lambda^{s+1}}{1-\lambda} \,.
\label{SNIydecay}
\end{equation}
Set $h(x) = \lambda^{-x} + \lambda^{x+1} -\lambda-1$ for $x=0,1,\dots$ and note that
\begin{align}
                        \lambda^{-(x+1)} &> \lambda^{x+1} \\
           \Leftrightarrow \lambda^{-(x+1)}(1-\lambda) &> \lambda^{x+1}(1-\lambda)\\
       \Leftrightarrow \lambda^{-(x+1)} - \lambda^{-x} &> \lambda^{x+1} -\lambda^{x+2} \\
\Leftrightarrow \lambda^{x+2}+\lambda^{-(x+1)}-\lambda -1 &> \lambda^{x+1}+\lambda^{-x}-\lambda -1
\end{align}
whence $h(x)$ is strictly monotonically increasing. Since $h(0)=0$ and
by setting $x= s-k$ it readily follows that
$$\lambda^{s+1}-\lambda^{-k+1}+\lambda^{s-2k+1}-\lambda^{-k} \le \lambda^{s+1}- \lambda^{-s}$$
which implies \eqref{SNIydecay} and hence
\begin{equation}
|\phi_k| \le D \lambda^{|k|} \frac{\lambda^{-s}}{1-\lambda}\,\,,\qquad\quad k \in {\mathbb Z}\,.
\end{equation}
\fi
In the next step we estimate $\|(S_N-S)S^{-1}_N P_N g_{m}\|$. Write $e^{(N)}_m = (S_N - S)
\phi^{(N)}$ and observe that $\|e^{(N)}_m\|^2 =\sum_{k=-\infty}^{\infty}|(e^{(N)}_m)_{k}|^2
=2\sum_{k=0}^{\infty}|(e^{(N)}_m)_{k}|^2$ (note that $(e^{(N)}_m)_{0} = 0$). For
$k=0,1.,\dots$ consider
\begin{equation}
(e^{(N)}_m)_{k} = \sum_{l=-\infty}^{\infty} (S_N - S)_{k,l} (\phi^{(N)}_m)_{l} =
\sum_{l=k-2s}^{k+2s} (S_N - S)_{k,l} (\phi^{(N)}_m)_{l}\notag
\end{equation}
Clearly $(e^{(N)}_m)_{k} = 0$ for $k \le N$, hence we only have to consider the
case $k > N$:
\begin{align}
|(e^{(N)}_m)_{k}| & = |\sum_{l=k-2s}^{k+2s} (S_N - S)_{k,l} (\phi^{(N)}_m)_{l} | \le
 B \frac{D \lambda^{-s}}{1-\lambda} \sum_{l=k-2s}^{k+2s} \lambda^{|l|} \notag \\
& \le \frac{B D \lambda^{-s}}{1-\lambda} \frac{\lambda^{k-2s}-\lambda^{k+2s}}{1-\lambda}
\le \frac{B D}{(1-\lambda)^2} \lambda^{k-3s}\,, \notag
\end{align}
whence
$$|(e^{(N)}_m)_{k}|^2 \le \left(\frac{BD}{(1-\lambda)^2}\right)^2 \lambda^{2k-6s}\,.$$
Due to $(S_N-S)_{k,l} = 0$ for $|k-l| > 2s$ and since
$(\phi^{(N)}_m)_{k} = 0$ for $|k|>N$ we obtain
$\sum_{k=0}^{\infty}|(e^{(N)}_m)_{k}|^2 = \sum_{k=N+1}^{N+2s}|(e^{(N)}_m)_{k}|^2$
and therefore
\begin{gather}
\|e^{(N)}_m\|^2 = 2\sum_{k=N+1}^{N+2s} |(e^{(N)}_m)_{k}|^2 \le
2 \frac{B^2 D^2}{(1-\lambda)^4} \lambda^{-6s}\sum_{k=N+1}^{N+2s} \lambda^{2k} \notag\\
\le \frac{2 B^2 D^2}{(1-\lambda)^4} \lambda^{2N}
    \frac{\lambda^{2-6s}-\lambda^{2-2s}}{1-\lambda^2}
\le \left(\sqrt{2} BD \lambda^{N} \frac{\lambda^{-3s}}{(1-\lambda)^3}\right)^2 \notag
\end{gather}
which implies
\begin{equation}
\|e^{(N)}_m\| \le \sqrt{2} BD \lambda^N (\lambda^{s}-\lambda^{s+1})^{-3}
\label{errorek}
\end{equation}
Combining~\eqref{errorek} with~\eqref{mainineq}
yields the bound~\eqref{gogondiff}.

The statements in (iii) and (iv) follow by applying
Proposition~\ref{proppolexpdecay} to~\eqref{mainineq}.
\end{proof}

\if 0
Further denote $E_N = S - S_N$ and observe that
\begin{equation}
(E_N)_{k,l} =
\begin{cases}
S_{k,l} & \text{if $|k|>N$ or $|l|>N$} \\
0       & \text{if $|k|\le 2s<N$ or $|l|\le 2s<N$} \\
0       & \text{if $|k-l| > 2s\,\,$ (since $S$ is $2s$-banded)}
\end{cases}
\label{Scond}
\end{equation}
Recall that $|\phi_k| \le D C_1 \lambda^{|k|}$ with
$C_1 = \frac{\lambda^{-s}}{1-\lambda}$.
We have
\begin{equation}
e^{(N)}_m_{k} = \sum_{l=-\infty}^{\infty} (E_N)_{k,l} \phi_l =
\sum_{l=k-2s}^{k+2s} (E_N)_{k,l} \phi_l \Rightarrow
|e^{(N)}_m_{k}| \le C_1 \sum_{l=-k-2s}^{k+2s} (E_N)_{k,l} \lambda^{|l|}
\label{}
\end{equation}
and
$\|e^{(N)}_m\|^2 =\sum_{k=-\infty}^{\infty}|e^{(N)}_m_{k}|^2 =2\sum_{k=0}^{\infty}|e_k|^2$
(note that $e^{(N)}_m_{0} = 0$).

We split this sum as follows
$$\sum_{k=0}^{\infty}|e^{(N)}_m_{k}|^2=\sum_{k=0}^{N-2s} |e^{(N)}_m_{k}|^2+
\sum_{k=N-2s+1}^{\infty} |e^{(N)}_m_{k}|^2$$
and consider each part separately.

Case $k\le N-2s$:
$$e^{(N)}_m_{k} =\sum_{l=k-2s}^{k+2s} (E_N)_{k,l}\phi_l =
              \sum_{l=-N}^{N} (E_N)_{k,l}\phi_l =0$$
due to the conditions listed in~\eqref{Scond}.

Case $k > N-2s$:
Due to the definition of the entries of $S$ in~\eqref{walnut} we can write
\begin{gather}
|e^{(N)}_m_{k}| \le D C_1 \sum_{l=k-2s}^{k+2s} |(E_N)_{k,l}| \lambda^{|l|}
=  C C_1 \frac{\lambda^{k-2s}-\lambda^{N+1}}{1-\lambda}
\le  C \frac{\lambda^{k-3s}}{(1-\lambda)^2}\,.
\end{gather}
It follows
\begin{equation}
|e^{(N)}_m_{k}|^2 \le \left(\frac{C}{(1-\lambda)^2}\right)^2(\lambda^{2k-6s})
\label{}
\end{equation}

\begin{remark}
In~\eqref{mainineq} we could have estimated
$\|(S_N - S) S^{-1}_N P_N g\|$ by $\|S_N - S\|\|S^{-1}_N P_N g\|$, this
however would provide an estimate of $\|\gamma - \gon\|$ which is much
too large to be of any use in applications. To obtain a meaningful
bound on the approximation error it is crucial to estimate
$\|(S_N - S) S^{-1}_N P_N g\|$ directly, as done in the proof above.
\end{remark}
\fi

\noindent
{\bf Remarks:}
\vspace*{-3mm}
\begin{itemize}
\setlength{\itemsep}{-0.5ex}
\setlength{\parsep}{-0.5ex}
\item[(i)] In spite of these results, good a priori estimates of the frame
bounds as given e.g.~in~\cite{Dau92,Jan97a} play an important role.
\item[(ii)] A different approach to approximate the inverse of the frame
operator by finite dimensional methods is due to Casazza and Christensen.
In~\cite{CC98} they consider a more general setting, however at the
cost that their estimates are less explicit and therefore less useful
in applications.
\end{itemize}

\section{Approximation of tight frames} \label{s:tight}

An important special case of frames are so-called {\em tight frames}, which
are defined in terms of equality of the upper and lower frame bound,
i.e., $A=B$. Besides the nice stability property $cond(S)=1$, tight frames are
distinguished by a simple expression for the duals, which are just scaled
versions of the original frame elements, more precisely one has
$\gamma_{m,n} = \frac{1}{A} g_{m,n}.$
Due to this close relationship to orthogonal bases the construction
of tight frames is of great interest in theory and in applications.
Daubechies~\cite{Dau90} has shown that for a given frame, say $\{\phi_k\}$
the set of functions $\{S^{-\frac{1}{2}}\psi_k\}$ constitutes a
tight frame. Recall that $S$ is positive definite, hence $S^{\frac{1}{2}}$ and $S^{-\frac{1}{2}}$
are uniquely determined.

Recall that $S^{\frac{1}{2}}$ commutes with the same operators
that commute with $S$. For shift-invariant systems this implies that
the tight frame generated by $\phi_{m,n} = S^{-\frac{1}{2}} g_{m,n}$
also constitutes a shift-invariant system. We want to approximate the
elements of the tight frame $\{\phi_{m,n}\} $ by the solutions of the finite
sections
$$(P_N S P_N)^{\frac{1}{2}} \phi^{(N)}_m = P_N g_{m}\,.$$
One can easily check that the entries of $S^{\frac{1}{2}}$ and $S^{-\frac{1}{2}}$ decay
exponentially off the diagonal, see e.g.~\cite{JM89,Lai94}.
The applicability of the finite section method follows now
from Theorem~5.1 in~\cite{GK94}. Along the same lines of the proof of
Theorem~\ref{theorem1} one can show that
$\phi^{(N)}_m := (P_N S P_N)^{-\frac{1}{2}} P_N g_{m}$ converges exponentially
fast to $\phi_m$. Since we do not have explicit expressions for the
decay of the entries of $S^{\frac{1}{2}}$ and $S^{-\frac{1}{2}}$ we only can give following
qualitative result for the approximation error.
\begin{theorem}
\label{theorem3}
Let $\{g_{m,n}\}$ be a shift-invariant frame for ${\lsp^2(\Zst})$ with
frame operator $S$ and frame bounds $A,B$. Assume that $\|g\|=1$ and
set $\phi_m = S^{-\frac{1}{2}} g_{m}$ and $\phi_m^{(N)} = (P_N S P_N)^{-\frac{1}{2}} P_N g$.
Then the finite section method is applicable to $S^{\frac{1}{2}}$. If
moreover $g$ is compactly supported with $g(k) = 0$ for $|k|>s$ then
$$\|\phi_m -\phi_m^{(N)} \| \le K \beta^{N} \qquad N=0,1,\dots$$
for $\beta \subset (0,1)$ and some constant $K>0$, where $\beta$ and $K$
depend on the frame bounds $A,B$ and the support length $s$, but
are independent of $N$.
\end{theorem}

\section{Rate of convergence by ``canonical'' approximation using a
periodic model}
\label{s:periodic}

We have seen in Section~\ref{s:rate} that the functions $\gamma_{m,n}^{(N)}$ which are
the duals to the truncated functions $P_N g_{m,n}$ converge exponentially fast
to the duals $\gamma_{m,n}$ of the infinite-dimensional problem.
So from a theoretical point of view we know that we can compute a good
approximation to $\gamma_{m}$  by solving the finite system $S_N \gamma_{m}^{(N)} = g_{m}^{(N)}$.
From a numerical point of view, however, we are interested in solving
$S_N \gamma_{m}^{(N)} = g_{m}^{(N)}$ as efficient as possible.

Unfortunately due to the
truncation of $S$ we loose some structural properties of the problem.
For instance the inverse of $S$ or the product $S^2$ still have
block Toeplitz structure, whereas $S^{-1}_N$ and $S_N^2$ are no longer
of Toeplitz type. Certainly one could apply one of the
fast solvers for Toeplitz systems, but we can also try to find a
more canonical way to design a finite dimensional model, which allows
to preserve the perfect symbol calculus of Laurent operators.
This is one motivation to use a periodic model for
the design of numerical algorithms.

Despite the fact that the periodic model is widely used in connection with
shift-invariant systems, almost no attempt has been made to
investigate the relation between the duals of the periodic finite model
and the duals of the infinite dimensional problem.

One exception is~\cite{Jan97a} where Janssen shows
that if the $g_{m,n}$ constitute a Gabor frame for ${\lsp^2(\Zst})$ with frame
bounds $A,B$ and if $g_{m,n} \in {\lisp(\Zst)}$, then the periodized functions
$g_{m,n}^{per}(k)= \sum_{j=-\infty}^{\infty}g_{m,n}(k-jL)$ constitute
a frame for ${\mathbb C}^L$ with frame bounds $A,B$.
Furthermore the duals for the periodized frame can be obtained by
periodizing the duals of the original frame elements. The drawback of
this otherwise very appealing result is that the length of the period
$L$ has to be the least common multiple of the time shift parameter $a$ and
the number of frequency channels $M$. This condition can be
a serious restriction in practice.  For instance take $a=2, M=3$, then
the periodized functions are of length 6, which may be much too short for
many applications.

In this section we will clarify the question under which conditions the
duals computed within the periodic model converge to the duals
$\gamma_{m,n}$ when the length of the period tends to infinity.

Let the functions $g_{m}$ be compactly supported, such that $g_{m}(k) = 0$
for $|k|>s$. We construct our periodic model, as usual, by
extending the truncated  functions $P_N g_{m}$ periodically
beyond the interval $[-N,N]$ for $N >s$ by setting
\begin{equation}
\label{cgm}
{^{\footnotesize P}\!\!g_{m}}(k+lN) := g_{m}(k) \qquad \text{for}\,\, k=-N,\dots,N,\,\,, l \in {\mathbb Z}\,.
\end{equation}
The other elements of our periodic finite shift-invariant system are now
given by
\begin{equation}
\label{cgmn}
{^{\footnotesize P}\!\!g_{m,n}}(k) = {^{\footnotesize P}\!\!g_{m}}(k-na)
\end{equation}
where the shift, according to our model, is understood as circulant shift.
The periodicity of the ${^{\footnotesize P}\!\!g_{m,n}}$ implies that the corresponding frame operator
is a block circulant matrix, rather than merely a block Toeplitz matrix.
Observe that the entries of ${^{\footnotesize P}\!S_N}$ deviate from the entries of $S_N$ only
in the lower right and upper left corner of the matrix.

In other words we have replaced the block Toeplitz matrix $S_N$ in
the system $S_N \gamma_{m}^{(N)} = g_{m}^{(N)}$ by a block circulant matrix.
However replacing a block Toeplitz system by a block circulant matrix
in a linear system of equations will in general significantly affect
the solution.

Moreover it is a priori not clear under which conditions invertibility
of $S_N$ implies invertibility of ${^{\footnotesize P}\!S_N}$. For instance take following
Hermitian positive definite Toeplitz matrix $T$ and its
``circulant completion'' ${^{\footnotesize P}T}$:
\begin{gather}
T =
\begin{bmatrix}
2  & -1 & 0  & 0  \\
-1 &  2 & -1 & 0  \\
0  & -1 &  2 & -1 \\
0  &  0 & -1 & 2  \\
\end{bmatrix}
\,, \qquad  {^{\footnotesize P}T} =
\begin{bmatrix}
2  & -1 & 0  & -1 \\
-1 &  2 & -1 & 0  \\
0  & -1 &  2 & -1 \\
-1 &  0 & -1 & 2  \\
\end{bmatrix}\notag
\end{gather}
then $T$ is invertible, whereas ${^{\footnotesize P}T}$ is obviously
not invertible.

\if 0
It is common practice is the mathematical and engineering community
to interpret a finite length signal $\{f(k)\}_{k=0}^{N-1}$ as infinite
periodic signal by extending the signal periodically at the boundaries,
(i.e., $f(lN+k)= f(k)$ for $l\in {\mathbb Z}, k=0,\dots,N-1$), instead of
padding it with zeros at the end points. And right so, since doing all
arithmetics in the ring ${\mathbb Z}_N$ is more than a convenient way
to allow operations such as convolution which may ask for values
$f(-1)$ which otherwise would not be defined. The nice thing is
that by this circular extension one obtains a locally compact abelian
group which in turn allows us to preserve important algebraic properties
of operators defined on other locally compact abelian groups such
as ${\mathbb R},{\mathbb Z},{\mathbb T},{\Rst^d},\dots$. For example Laurent operators
on ${\lsp^2(\Zst})$ commute with one another, however finite (or
even one-sided infinite) sections, i.e. finite Toeplitz matrices
do not commute. But circulant matrices (``periodic'' Toeplitz
matrices) preserve the commutation property. Moreover we can make use of
the abstract results in Fourier analysis~\cite{Rud62} or time-frequency
analysis~\cite{Hol95,FK98,Gro98,TA98} on locally compact abelian groups.

We have seen in Section~\ref{s:rate} that the approximate dual frames
obtained by using the finite model arising from truncating the involved
functions to the interval $[-N,N]$
converge exponentially fast to the original duals. It is a priori not clear
how the solution will change, if we extend those truncated functions
periodically across the boundaries. This periodic extension implies that
the resulting ``frame operator'' is a block circulant matrix, rather than
merely a block Toeplitz matrix. However replacing in a block Toeplitz
system $Tx=y$ the matrix $T$ by a block circulant matrix
will in general significantly affect the solution.
Thus the question we want to answer in this section is, do the
``dual'' functions computed by using a periodic model still converge
to the original duals, and if yes, what can we say about the speed
of convergence.
\fi

The following theorem justifies the usage of the circulant model for
the approximation of dual shift-invariant systems. Implicitly the theorem
also provides an estimate of the error we make when we
approximate the solution of a block Toeplitz system by solving the
related block circulant system.
\begin{theorem}
\label{theorem2}
Let $\{g_{m,n}\}$ be a shift-invariant frame for ${\lsp^2(\Zst})$ with
frame bounds $A,B$. Assume that $\|g_{m}\|=1$ and that the $g_{m}$ are compactly
supported with $g_{m}(k) = 0$ for $|k|>s$. Let $N>2s$ and let ${^{\footnotesize P}\!\!g_{m,n}}$
be the periodized functions as defined in~\eqref{cgm} and~\eqref{cgmn} with
frame operator ${^{\footnotesize P}\!S_N}$. Then ${^{\footnotesize P}\!S_N}$ is invertible and ${^{\footnotesize P}\!\!\gamma_{m}^{(N)}} = {^{\footnotesize P}\!S_{N}^{-1}} {^{\footnotesize P}\!\!g_{m}}$
converges to $\gamma_{m}$ for $N \rightarrow \infty$. If $N>3s$ then the rate
of convergence can be estimated by
\begin{equation}
\|P_N \gamma_{m} -{^{\footnotesize P}\!\!\gamma_{m}^{(N)}} \| \le  3 \sqrt{2} C \lambda^N (\lambda^{s}-\lambda^{s+1})^{-3}
\label{gogondiff2}
\end{equation}
where $\kappa = \frac{B}{A}, q = \frac{\sqrt{\kappa} -1}{\sqrt{\kappa}+1}$,
$\lambda = q^{\frac{1}{2s}}$ and
\begin{equation}
\label{constantdef1}
C = \frac{1}{A} \max\{2\kappa,(1+\sqrt{\kappa})^2 \}
\end{equation}
\end{theorem}

\begin{proof}
First we show that invertibility of $S$ implies invertibility of ${^{\footnotesize P}\!S_N}$.
The results about the spectrum of block circulant matrices in the proof
of Theorem~3 in~\cite{ZTA97} in combination with Corollary~1
in~\cite{ZTA97} imply that $$\sigma({^{\footnotesize P}\!S_N}) \subseteq \sigma(S)\,.$$
$\sigma(S)$ is bounded away from zero since $A>0$, hence
it follows that ${^{\footnotesize P}\!S_N}$ is invertible.

To prove~\eqref{gogondiff2} we write
\begin{equation}
\|P_N \gamma_{m} - {^{\footnotesize P}\!\!\gamma_{m}^{(N)}} \| \le
 \|\gamma_{m} - \gamma_{m}^{(N)} \|  + \|P_N \gamma_{m}^{(N)} -{^{\footnotesize P}\!\!\gamma_{m}^{(N)}}\|\,.
\label{circdiff}
\end{equation}
We already have estimated $\|\gamma_{m} - \gamma_{m}^{(N)} \|$ in~\eqref{gogondiff} of
Theorem~\ref{theorem1}, so it remains to estimate $\|P_N \gamma_{m}^{(N)} -{^{\footnotesize P}\!\!\gamma_{m}^{(N)}}\|$.
Proceeding analogously to
the calculation preceding equation~\eqref{mainineq} we can write
\begin{equation}
\|\gamma_{m}^{(N)} - {^{\footnotesize P}\!\!\gamma_{m}^{(N)}}\| \le A^{-1} \|({^{\footnotesize P}\!S_N} - S_N) {^{\footnotesize P}\!S_{N}^{-1}} P_N g_{m}\|\,.
\label{mainineq2}
\end{equation}
Note that ${^{\footnotesize P}\!S_N}$ is three-band
matrix, with one band centered at the main diagonal, and two other bands of
width $2s$ located at the lower left and upper right corner of the matrix.
It follows from Proposition~5.1 in~\cite{DMS84} that the entries of
${^{\footnotesize P}\!S_{N}^{-1}}$ decay exponentially off the diagonal and off the lower right and
upper left corner. More precisely
\begin{equation}
|({^{\footnotesize P}\!S_{N}^{-1}})_{k,l}| \le
\begin{cases}
D \lambda^{|k-l|} & \text{if $0 \le |k-l| \le N$} \\
D \lambda^{2N+1-|k-l|} & \text{if $N+1 \le |k-l| \le 2N$}
\end{cases}
\label{Scond}
\end{equation}
where $D$ and $\lambda$ are as in Theorem~\ref{thdemko} with $m=2s$.

Before we show that $\phi^{(N)}_m: = {^{\footnotesize P}\!S_{N}^{-1}} {^{\footnotesize P}\!\!g_{m}}$ also decays exponentially,
observe that due to the construction of ${^{\footnotesize P}\!S_N}$ the matrix
$E_N: = {^{\footnotesize P}\!S_N} - S_N$ is a sparse matrix having non-zero entries only
in the upper right and the lower left corner, as illustrated in~\eqref{CNSNmat}
\begin{gather}
\qquad \qquad \begin{matrix}
& & & & & \text{\tiny{$N\!-\!2s+1$}} & & & \text{\tiny{$N$}} \notag
\vspace*{-6mm}
\end{matrix} \\
E_N =
\begin{bmatrix}
   & & &           & \times       & \dots  & \times       \\
   & & &           &         & \ddots & \vdots  \\
   & & &           &         &        & \times       \\
   & & & \text{\Large{$0$}}  &         &        &         \\
 \times & & &           &         &        &         \\
\vdots & \ddots & & &        &        &         \\
\times & \dots & \times &    &         &        &         \,\,\,.
\label{CNSNmat}
\end{bmatrix}
\end{gather}
Denoting $e^{(N)}_m = E_N {^{\footnotesize P}\!S_{N}^{-1}} {^{\footnotesize P}\!\!g_{m}}$, it readily follows from the sparsity
of $E_N$ that only the first $2s$ and the last $2s$ components of
$e_N$ are non-zero, hence only the decay of the first $2s$ and last $2s$
components of $\phi^{(N)}_m = {^{\footnotesize P}\!S_{N}^{-1}} {^{\footnotesize P}\!\!g_{m}}$ is of interest.

Since ${^{\footnotesize P}\!S_N}$ is a three-band matrix, the decay behavior of the entries
of ${^{\footnotesize P}\!S_N}^{-1}$ is a little bit more complicated than for ordinary band
matrices. We split the analysis of the decay of the entries of $\phi^{(N)}_m$ into
two steps by considering first the entries $(\phi^{(N)}_m)_{k}$ for
$k = N-2s,\dots,N-m-1$ and then those for $k=N-m,\dots,N-1$.

\noindent
Case $k=N-2s+1,\dots,N-s$: \\
$$(\phi^{(N)}_m)_{k} = \sum_{l=-N}^{N} ({^{\footnotesize P}\!S_{N}^{-1}})_{k,l} ({^{\footnotesize P}\!\!g_{m}})_l =
\sum_{l=-s}^{s} ({^{\footnotesize P}\!S_{N}^{-1}})_{k,l}({^{\footnotesize P}\!\!g_{m}})_l$$ due to the
compact support of $g_{m}$. Hence
\begin{equation}
|(\phi^{(N)}_m)_{k}| \le |\sum_{l=-s}^{s}({^{\footnotesize P}\!S_{N}^{-1}})_{k,l}|
\le D \sum_{l=-s}^{s} \lambda^{|k-l|}  \,.
\label{}
\end{equation}
The assumption $N > 3s$ implies that $k \ge l$ hence
\begin{align}
D \sum_{l=-s}^{s} \lambda^{|k-l|} = D \sum_{l=-s}^{s} \lambda^{k-l}
= D \lambda^{k} \frac{\lambda^{-s}-\lambda^{s+1}}{1-\lambda}
\le \frac{D}{1-\lambda}\lambda^{k-s}
\end{align}

\noindent
Case $k=N-m,\dots,N-1$: \\
\begin{align}
|(\phi^{(N)}_m)_{k}| & \le |\sum_{l=-s}^{s}({^{\footnotesize P}\!S_{N}^{-1}})_{k,l}|
\le \sum_{l=-s}^{k-N-1}|({^{\footnotesize P}\!S_{N}^{-1}})_{k,l}| +
\sum_{l=k-N}^{s}|({^{\footnotesize P}\!S_{N}^{-1}})_{k,l}| \notag \\
& \le D \sum_{l=-s}^{k-N-1} \lambda^{2N+1-k+l}
+ D \sum_{l=k-N}^{s} \lambda^{k-l} \notag \\
& = D \lambda^{2N+1-k-s} \sum_{l=0}^{k-N+s-1} \lambda^{l}
+ \frac{D}{1-\lambda} \lambda^{k} \frac{\lambda^{-s}-\lambda^{-(k-N-1)}}{1-\lambda}\notag \\
& =\frac{D}{1-\lambda}
\left(\lambda^{2N+1-k-s}-\lambda^{-N+1}+\lambda^{k-s}-\lambda^{N+1}\right) \notag \\
& \le \frac{D}{1-\lambda}(\lambda^{k-s}+\lambda^{2N+1-k-s}) \le
\frac{2D}{1-\lambda}\lambda^{k-s}
\end{align}
Note that
$\|e^{(N)}_m\|^2 = \sum_{k=-N}^{N}|(e^{(N)}_m)_{k}|^2 = 2\sum_{k=N-2s+1}^{N}|(e^{(N)}_m)_{k}|^2$.
Due to the special sparsity structure of $E_N$ as illustrated
in~\eqref{CNSNmat} we can write
\begin{align}
|(e^{(N)}_m)_{k}| & \le \sum_{l=2N-2s+1+k}^{N} |(E_N)_{k,l}| |\phi_l|
\le \frac{2BD}{1-\lambda} \lambda^{-s} \sum_{l=2N-2s+1+k}^{N} \lambda^l \notag \\
& \le \frac{2BD}{(1-\lambda)^2} \lambda^{-s} (\lambda^{2N-2s+1+k} - \lambda{N+1}
\le \frac{2BD}{(1-\lambda)^2} \lambda^{k-3s+2N} \notag
\end{align}
It follows
$$|(e^{(N)}_m)_{k}|^2 \le \left(\frac{2BD}{(1-\lambda)^2}\right)^2 \lambda^{2k+4N-6s}$$
and
\begin{gather}
\|e^{(N)}_m\|^2  =  2 \sum_{k=-N}^{-N+2s-1} |(e^{(N)}_m)_{k}|^2
\le 2 \frac{(2BD)^2}{(1-\lambda)^4} \sum_{k=-N}^{-N+2s-1}
\lambda^{2k-6s+4N} \notag \\
\le 2 \frac{(2BD)^2}{(1-\lambda)^4}
\frac{\lambda^{2N-6s} - \lambda^{2N-2s}}{1-\lambda}
\le \left(2 \sqrt{2} \frac{BD}{(1-\lambda)^3} \lambda^{N-3s}\right)^2 \notag
\end{gather}
and therefore
$$
\|e^{(N)}_m\| \le 2 \sqrt{2} \frac{BD \lambda^{-3s}}{(1-\lambda)^3} \lambda^{N}
= 2 \sqrt{2} BD (\lambda^{s}-\lambda^{s+1})^{-3} \lambda^{N}
$$
which together with~\eqref{mainineq2} yields the desired
estimate~\eqref{gogondiff2}.
\end{proof}

The corresponding variant of Theorem~\ref{theorem3} for the approximation
of a tight frame via the periodic model is left to the reader.

In fact, the periodic extension of finite signals is more than just a
simple and convenient way to hanlde boundary problems. It is in some sense
the ``canonical'' way to set up a discrete model, when translations
come into play, since it allows to preserve important mathematical
properties of the infinite dimensional problem.

More generally speaking one preserves the underlying group structure
of the original problem. By doing so, we can immediately apply the
abstract results derived for shift-invariant systems~\cite{FK98,Gro98,TA98}.
In fact, sometimes, (e.g.~in the case of
Gabor analysis~\cite{Str98}) we gain even more structural properties compared
to the infinite dimensional problem, due to the finiteness of the
underlying abelian group.

\if 0
\section{Approximation of tight frames} \label{s:tight}

An important special case of frames are so-called {\em tight frames}, which
are defined in terms of equality of the upper and lower bound, i.e., $A=B$.
Besides the nice stability property $cond(S)=1$, tight frames are
distinguished by a simple rule for the duals, which are just scaled
versions of the original frame elements, more precisely one has
$\gamma_{m,n} = 1/A g_{m,n}.$
Due to this close relationship to orthogonal bases the construction
of tight frames is of great interest in theory and in applications.
Daubechies~\cite{Dau90} has shown that for a given frame, say $\{\phi_k\}$
the set of functions $\{S^{-\frac{1}{2}}\psi_k\}$ constitutes a
tight frame. Recall that $S$ is positive definite, hence $S^{\frac{1}{2}}$ and $S^{-\frac{1}{2}}$
are uniquely determined.

Recall that the square root of $S$ commutes with the same operators
that commute with $S$. For shift-invariant systems this implies that
the tight frame generated by $\phi_{m,n} = S^{-\frac{1}{2}} g_{m,n}$
also constitutes a shift-invariant system. We want to approximate the
elements of the tight frame $\{\phi_{m,n}\} $ by the solutions of the finite
sections
$$(P_N S P_N)^{\frac{1}{2}} \phi^{(N)}_m = P_N g_{m}\,.$$
One can easily check that the entries of $S^{\frac{1}{2}}$ and $S^{-\frac{1}{2}}$ decay
exponentially off the diagonal, see e.g.~\cite{JM89,Lai94}.
The applicability of the finite section method follows now
from Theorem~5.1 in~\cite{GK94}. Along the same lines of the proof of
Theorem~\ref{theorem1} one can show that
$\phi^{(N)}_m := (P_N S P_N)^{-\frac{1}{2}} P_N g_{m}$ converges exponentially
fast to $\phi_m$. Since we do not have explicit expressions for the
decay of the entries of $S^{\frac{1}{2}}$ and $S^{-\frac{1}{2}}$ and thus for the speed
of convergence, we only can give following qualitative result.
\begin{theorem}
\label{theorem3}
Let $\{g_{m,n}\}$ as defined in~\eqref{sis} be a frame for ${\lsp^2(\Zst})$ with
frame operator $S$ and frame bounds $A,B$. Assume that $\|g\|=1$ and
set $\phi_m = S^{-\frac{1}{2}} g_{m}$ and $\phi_m^{(N)} = (P_N S P_N)^{-\frac{1}{2}} P_N g$.
Then the finite section method is applicable to $S^{\frac{1}{2}}$. If
moreover $g$ is compactly supported with $g(k) = 0$ for $|k|>s$ then
$$\|\phi_m -\phi_m^{(N)} \| \le K \beta^{N} \qquad N=0,1,\dots$$
for $\beta \subset (0,1)$ and some constant $K>0$, where $\beta$ and $K$
depend on the frame bounds $A,B$ and the support length $s$, but
are independent of $N$.
\end{theorem}

\subsection{Some remarks on the computation of compactly supported
dual Gabor frames} \label{ss:}

In~\cite{Boe97} H.~B\"olcskei has investigated under which
conditions the minimal norm dual Gabor window $\gamma$ is compactly supported
for given compactly supported window $g$. It turns out that
these conditions are rather restrictive. In general we cannot
expect that the minimal norm dual has compact support.
But in case of oversampled Gabor transforms there are several choices
for dual windows and it is well-known that this freedom can be used
to design compactly supported dual windows. This freedom is also
used in the approach of Qian and Chen to compute dual windows
for Gabor expansions in case of long signals~\cite{QC96}.

In the sequel we briefly describe the shortcomings of the method
proposed by Qian and Chen and show how these shortcomings
can be easily overcome by making use of Theorem~\ref{theorem2}.
\fi

\section*{Acknowledgement}
I want to thank B. Silbermann and G. Zimmermann for their valuable
suggestions.

The author has been partially supported by Schr\"odinger scholarship
J01388-MAT of the Austrian Science foundation FWF.

\bibliographystyle{plain}

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Our debut! The first animation in an explainer video we are currently working on at Motion Authors. View an extended version together with sound at https://vimeo.com/73838888.
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