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

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, The University of Iowa:

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, The University of Iowa:

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.

Wavelets through a Looking Glass: The World of the Spectrum