**Workshop on ****wavelet analysis, fractal
geometry, iterated function systems and applications**

**
June 13, 2007**** **

**
Centre
for Mathematical Sciences**

**Lund**** university**

**Sweden**** **

**Organized
by: **

sergei.silvestrov@math.lth.se

http://www.maths.lth.se/matematiklth/personal/ssilvest/

http://www.maths.lth.se/matematiklth/personal/ssilvest/nangseminar.html

**Supported by: **

Crafoord Foundation and STINT Foundation

**Program**

Wednesday,
June 13, 10.15-11:05 in MH:333:

*Noncommutative** geometry och
noncommutative analysis seminar*

**
Örjan Stenflo**,
Uppsala University:

Random V-variable fractals

Wednesday,
June 13, 11.10-12:00 in MH:333:

*Noncommutative** geometry och
noncommutative analysis seminar*

**
Ole Christensen**, Danmarks Tekniske Universitet:

Gabor systems and frames.

12:00 -- 13.00 Lunch

Wednesday,
June 13, kl 13.15-15:00
in MH:333:

*Noncommutative** geometry och
noncommutative analysis seminar*

**Palle E. T. Jorgensen**,

Wavelets, fractals,
iterated function systems and operator representations.

**Abstracts**

**
Örjan Stenflo**,
Uppsala University:

**Title:** Random V-variable fractals

**Abstract: **

A standard way to generate probability measures supported on fractal sets is to regard them as limiting probability distributions for processes obtained from random iterations of functions. It is typically not possible to generate natural random fractals like e.g. Brownian motion in a similar way due to the complexity of these objects. This has restricted applications in fractal modeling. In joint work with Michael Barnsley and John Hutchinson we introduced V-variable fractals as a way of resolving this. Random V-variable fractals can be generated quickly as "points" along trajectories of a fractal-valued random iteration process. Simultaneously they can be used to approximate "standard" random fractals.

**
Ole Christensen**, Danmarks Tekniske Universitet:

**Title:** Gabor systems and frames.

**Abstract: **In signal processing and pure
mathematics it is useful to represent complicated
signals or functions f in terms of linear
combinations of simpler building blocks ek;
the classical case, where ek
is assumed to be an orthonormal basis for a Hilbert
space H to which the relevant signals belong, leads to
the representation f=
Σ< f,ek>
ek.
However, the assumption that ek
forms an ONB is quite restrictive, and it limits the other
properties one can expect from ek.
We discuss concrete cases from wavelet analysis and
Gabor theory, where desirable properties of ek
can not be combined with ek
being an ONB. It turns out that much more freedom can
be attained by assuming that ek
is a frame rather than an ONB. We will discuss recent
results concerning frames, with particular focus on
Gabor frames.

**Palle E. T. Jorgensen**,

**Title: **
Wavelets, fractals,
iterated function systems and operator representations.

**Abstra****ct:**

The lectures, combining analysis and tools from mathematical probability, give a systematic presentation of recent trends in three fields, wavelets, signals and fractals. The unity of basis constructions and their expansions is emphasized as the starting point for the development of bases that are computationally efficient for use in several areas from wavelets to fractals.

From operator algebra theory, a key topic in the talks will be representations of the Cuntz algebras.

They aim to bring together tools from engineering and math, especially from signal- and image processing, and from harmonic analysis and operator theory. The presentation is aimed at graduate students.

Special:

• motivation;

• glossary of terms, their use in mathematics and in engineering;

• graphic renditions of algorithms, and separate illustrations;

• explaining engineering terms to mathematicians, and operator theory to engineers;

• guide to the literature.

Palle E.T. Jorgensen is a Professor of Mathematics at the University of Iowa.

He has taught some of this in courses over the last several years.

He is the author of a GTM Springer book v 234 in 2006.

**
Analysis and Probability: Wavelets, Signals,
Fractals, **
Graduate Texts in Mathematics, Vol. 234,
Springer**.**

*Analysis and Probability: Wavelets, Signals, Fractals*

And his other most recent book was written jointly with Ola Bratteli and is entitled

*
Wavelets through a Looking Glass*, Applied and Numerical
Harmonic Analysis,
Birkhäuser, 2002.