WAVELETS AND APPLICATIONS
Barcelona, July 1-6, 2002

Wavelets 2002 The Institute of Mathematics of the University of Barcelona (IMUB) organizes a workshop on Wavelets and Applications, July 1-6, 2002 (Barcelona). The Organizing Committee is: María J. Carro, Joan Cerdà, Joaquim Martín, and Javier Soria.

The course is addressed to graduate students and young researchers not necessarily specialized on wavelets.

IMUB

There will be four main basic courses:

- Introduction and basic aspects of wavelets theory, G. Weiss (PDF ).

- Wavelets and probability, R. Gundy (PDF ).

- Wavelets and numerical methods, C. Canuto and A. Tabacco (PDF, Slides 1, Slides 2, Slides 3 ).

- Computer-based wavelet analysis, T. Nguyen (PDF ).

The workshop will also consist of several specialized lectures related and complementing the above courses. (The workshop Lecture Notes, published by the IMUB, can be downloaded in PDF format.)

The lectures will take place at the Faculty of Mathematics building, located at the downtown campus of the University of Barcelona.

The registration fee is 90 €   (EUROS). Due to restrictions in the use of computer laboratories, the number of participants is limited (there will be two laboratory groups, each having two different sessions. See the Programme for more details). The inscription will be done by rigorous date of payment.

With the support of the Spanish Mathematical Society (RSME), the organizers will offer a few grants to cover some of the participants expenses. This money will be awarded directly by the RSME after the workshop.

On Thursday, July 4, there will be an organized tour through Barcelona. This activity is supported by the City Council and it is free for the participants.

On Friday, July 5, we will have a special dinner on the Escola d'Hostaleria de Barcelona.

If you need help for booking your hotel room, check the information available at the ACCOMMODATION FORM. Please address all questions you may have on this matter to the Congress Office of the University of Barcelona.

To obtain more information about the city of Barcelona you may want to visit the web site of the City Council or the web site of Tourism in Barcelona. You can also consult an interactive map of Barcelona or check the weather forecast.


PROGRAMME


MONDAY, July 1

9:00-10:30 Reception

10:30-11:00 Welcoming address

11:00-12:00 Introduction and basic aspects of wavelets theory (1)

12:00-13:00 Wavelets and numerical methods (1)

13:00-15:00 Lunch

15:00-16:00 Wavelets and probability (1)

16:00-17:00 Eugenio Hernández (U. Autonóma de Madrid): Non-linear approximation and wavelets (PDF )

TUESDAY, July 2

9:30-10:30 Computer-based wavelet analysis

10:30-11:30 Wavelets and probability (2)

11:30-12:00 Break

12:00-13:00 Introduction and basic aspects of wavelets theory (2)

13:00-15:00 Lunch

15:00-16:00 Wavelets and numerical methods (2)

16:00-17:00 Miquel Ralló (Laboratori de Processat d'Imatges, UPC): Wavelet based techniques for textile inspection (PDF )

WEDNESDAY, July 3

9:00-10:30 Computer-based wavelet analysis (Lab. session 1)

10:30-11:00 Break

11:00-12:00 Wavelets and probability (3)

12:00-13:00 Introduction and basic aspects of wavelets theory (3)

13:00-15:00 Lunch

15:00-16:00 Wavelets and numerical methods (3)

16:15-17:45 Computer-based wavelet analysis (Lab. session 1)

THURSDAY, July 4

9:30-10:30 Edward Wilson (Washington U.): The algebra of shift invariant spaces and applications to wavelet theory (PDF )

10:30-11:30 Hrvoje Sikic (U. Zagreb): Tight frame wavelets and the dimension function (PDF )

11:30-12:00 Break

12:00-13:00 Wavelets and numerical methods (4)

13:00-15:00 Lunch

15:00-18:00 Organized tour through Barcelona

FRIDAY, July 5

9:00-10:30 Computer-based wavelet analysis (Lab. session 2)

10:30-11:00 Break

11:00-12:00 Wavelets and probability (4)

12:00-13:00 Introduction and basic aspects of wavelets theory (4)

13:00-15:00 Lunch

15:00-16:00 Wavelets and numerical methods (5)

16:15-17:45 Computer-based wavelet analysis (Lab. session 2)

21:00-23:00 Special dinner

SATURDAY, July 6

9:30-10:30 Claudio Canuto (Politecnico di Torino): Wavelet-based analysis of singular perturbation problems (PDF )

10:30-11:30 Wavelets and probability (5)

11:30-12:00 Break

12:00-13:00 Introduction and basic aspects of wavelets theory (5)


ABSTRACTS OF THE COURSES


- Introduction and basic aspects of wavelets theory (Guido Weiss , Washington University) (PDF ).

We begin with a general description of reproducing systems on n dimensional Euclidean space; this will include both the continuous and discrete cases. We then will proceed to give a unified approach for the study of all these collections. The first goal will be to present a theorem that characterizes all such systems. Particular special cases are wavelets (in all dimensions, finitely generated, with very general dilations), Gabor systems and Gabor-like systems, the frame versions of these systems and others that are more general. Many concrete examples will be constructed. Other topics that are related to this that will be discussed are: shift invariant subspaces, connectivity properties of these systems, oversampling.

  1. A bit of history of wavelets and related systems and why they arose.
  2. A first lecture on the characterization of continuous and discrete affine (wavelet) and related systems.
  3. A second lecture on these characterizations.
  4. The existence of these systems.
  5. Some of the basic properties of these systems.

- Wavelets and probability (Richard Gundy, Rutgers University) (PDF ).

In these lectures, we want to illustrate some connections between wavelet analysis and probability. The emphasis will be in the direction from probability to wavelet analysis, rather than the other way. At this point, we have two principal topics in mind:

  1. Oscillation and convergence of wavelet series, trigonometrical series, and the clues we have from the study of Brownian motion, harmonic functions, and a class of martingales. Konyagin's (relatively) recent solution to a problem of Menchov for trigonometric sums will be discussed in relation to the same problem for wavelet series. A theorem on "local linear independence" of scaling functions, due to Yves Meyer, turns out to provide a substitute for stopping time arguments.

  2. The Markov process associated with a scaling function. Which periodic functions are low-pass filters for some scaling function? Necessary and sufficient conditions have been given by numerous people under special conditions. Most notably, A.Cohen provided the first solution when the candidate function is a trigonometric polynomial. For a while, it was thought that his conditions were necessary and sufficient under the assumption that the function is simply continuous. This, it turns out, is not the case. A completely general criterion exists, and is expressed by using the function to create an interesting Markov process. A crucial step in the proof uses a version of Liouville's theorem, the classical fact that in R^n, the only bounded harmonic functions are constants.

- Wavelets and numerical methods (Claudio Canuto and Anita Tabacco, Politecnico di Torino) (PDF, Slides 1, Slides 2, Slides 3 ).

  1. Compactly supported wavelets on the real line.
  2. Wavelets on the interval and boundary conditions.
  3. The Wavelet Element Method.
  4. Numerical approximation of elliptic equations.
  5. Nonlinear approximation and adaptivity.
- Computer-based wavelet analysis (Truong Nguyen, University of California, San Diego) (PDF ).

The theory, design and application of wavelets will be discussed in the course with emphasis on computer-based laboratory use. Topics to be discussed and demonstrated are multiresolution analysis; design of different wavelet types; image compression and signal denoising. The course consists of a lecture session followed by two hand-on PC-based laboratory sessions.


Speakers D. Gundy, C. Canuto, E. Wilson, G. Weiss, H. Sikic

E. Hernández, A. Tabacco, T. Nguyen

List of participants


IMUB
University of Barcelona
Gran Via 585
08071 Barcelona, Spain
Fax: 34-93 403 5963

wavelets@imub.ub.es


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