Workshop

Mathematical Analysis andApplications

In occassion of the retirement of Prof. Dr. Rupert Lasser

Neuherberg, Germany,

September 19 – 20, 2013

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Organizers:Wolfgang zu CastellMessoud EfendiyevFrank Filbir

Acknowledgement:The organizers would like to thank the Helmholtz Zentrum Munchen and theInstitute of Computational Biology for financial support.

We are especially indebted to Sandra Mayer for her valuable help during theorganization of the meeting.

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Contents

1 Aim 4

2 Workshop Location 5

3 List of Speakers 7

4 Abstracts of Talks 8

5 Scientific Program 13

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1 Aim

On the occasion of the retirement of Prof.Dr. Rupert Lasser the Helmholtz ZentrumMunchen will organize a workshop on

Mathematical Analysis andApplications

Rupert Lasser was the director of the In-

stitute of Biomathematics and Biometry at

Helmholtz Zentrum Munchen and had the

chair for Mathematics in Medicine and Life

Sciences at Technische Universitat Munchen

since 1997. In his work Rupert Lasser always

expressed his strong belief of the unity of pure

and applied mathematics. Since May 2013

Rupert Lasser is retired.

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2 Workshop Location

The workshop takes place at the campus of the Helmholtz Zentrum Munchen.

Address:Helmholtz Zentrum MunchenIngolstadter Landstraße 1D-85764 Neuherberg,Germany

Location:”Großer Horsaal”, building 33, room 106.

Directions:Information on how to get to the Helmholtz campus can be found onhttp://www.helmholtz-muenchen.de/ueber-uns/standorte/index.html#con1

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Campus Map

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3 List of Speakers

Hermann EberlDepartment of Mathematics and Statistics, University of Guelph, Ontario,Canada

Hans Georg FeichtingerInstitute of Mathematics, University of Vienna, Austria

Brigitte Forster-HeinleinInstitute of Applied Mathematics, University of Passau, Germany

Hartmut FuhrChair of Mathematics A, RWTH Aachen, Germany

Karlheinz GrochenigInstitute of Mathematics, University of Vienna, Austria

Francois HamelLATP, Faculte des Sciences et Techniques, Universite Aix-Marseille III, France

Eberhard KaniuthInstitute of Mathematics, University of Paderborn, Germany

Volkmar LiebscherInstitut fur Mathematik und Informatik, Universitat Greifswald, Germany

Hrushikesh N. MhaskarCalifornia Institute of Mathematics, andClaremont Graduate University, Claremont, U.S.A.

Jurgen PrestinInstitute of Mathematics, University of Lubeck, Germany

Holger RauhutChair of Mathematics C, RWTH Aachen, Germany

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4 Abstracts of Talks

• Cross-diffusion in biofilms.

Hermann Eberl, University of Guelph

Bacterial biofilms have been characterized as both, mechanical objects and

as spatially structured microbial populations. For dual species systems we

show that both view points lead to the same nonlinear cross-diffusion model.

We illustrate the role of cross-diffusion effects in preliminary simulations and

point out mathematical challenges for future research.

• Ideas towards postmodern analysis.

Hans Georg Feichtinger, University of Vienna

An occasion like this is a good moment to look back, review the changesin our common scientific field, i.e. Harmonic Analysis to see how it haschanged, and what the topics have been at the beginning and now in thelate phase of our career.

Since it was the time of Modern Analysis when we started I don’t know abetter word for the upcoming area “Postmodern Harmonic Analysis”. Af-ter the phase of abstraction (LCA groups, distributions) and diversification(FFT, computational HA, wavelets, Gabor) and many interesting applica-tions (e.g. in mobile communication or signal processing) it is time to thinkof a more integrative view-point, and bring the established theoretical bodycloser to applications, maybe also in the way how we teach Fourier Analysis.

I think this way of thinking is also very familiar to Rupert Lasser who was

all his academic life moving between the two poles.

• Wavelet coorbit theory in higher dimensions.

Hartmut Fuhr, RWTH Aachen

Coorbit theory provides a functional-analytic framework for the construction

and study of Banach frames arising from the action of an integrable represen-

tation. This talk is concerned with existence and basic properties of coorbit

spaces associated to wavelet transforms arising from an irreducible, square-

integrable representation of a semidirect product of the type G = Rd o H

acting naturally on L2(Rd). Here H is a suitably chosen, closed matrix

group. The talk provides a unified and rather general approach to a setting

that so far has only been studied for very special choices of affine group ac-

tions (such as the similitude group, or the shearlet group). It establishes the

well-definedness of a scale of Besov-type coorbit spaces, and provides the ex-

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istence of atomic decompositions for these spaces in terms of suitably chosen

band-limited Schwartz functions. Under suitable assumptions on the dual

action of H I establish easily verified concrete conditions for frame atoms,

in terms of vanishing moments, smoothness and decay. In particular, these

results imply the existence of compactly supported smooth atoms.

• Phase in signals and images.

Brigitte Forster-Heinlein, University of Passau

For many imaging applications the dogma ”images are real-valued” still in-

fluences the choice of the applied image processing methods. We give several

mathematical reasons to consider a more general class of transforms in that

context, i.e., complex-valued transforms that extract phase information. We

show that in combination with multiresolution methods they yield powerful

image processing methods for 1D, 2D and also for higher dimensional image

data. Applications show the performance of our concept.

• Wiener’s Lemma in Banach algebras and norm-controlledinversion.

Karlheinz Grochenig, University of Vienna

Wiener’s Lemma states that the inverse of an absolutely convergent Fourier

series without zeros is again an absolutely convergent Fourier series. We will

discuss several versions of Wiener’s Lemma in (non-commutative) Banach

algebras, for instance for algebras of matrices with off-diagonal decay, for

convolution operators on discrete nilpotent groups, or for pseudodifferen-

tial operators. We then study the question which of these algebras possess

normcontrol. While the algebra of absolutely convergent Fourier series does

not admit norm-control by a result of Nikolski, many algebras that are con-

structed with methods from approximation theory do admit normcontrol.

Many of these topics were started during my time at the former IBB.

• Inside structure of range-expanding populations.

Francois Hamel, Universite Aix-Marseille III

This talk will be focused on some mathematical aspects of a model for gene

surfing along an invasion front. This model describes the dynamics of com-

ponents inside a front. From a mathematical point of view, it corresponds

to a reaction-diffusion equation with a forced speed. I will discuss the case

of monostable, bistable or ignition reactions. In the monostable case, the

fronts are classified as pulled or pushed ones, depending on the propaga-

tion speed. It will be shown that any localized component of a pulled front

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converges locally to 0 at large time in the moving frame of the front, while

any component of a monostable pushed, bistable or ignition front converges

to a well determined positive proportion of the front. The results give a

more complete interpretation of the pulled/pushed terminology, which can

be extended to the case of general transition waves. This talk is based on

some joint works with J. Garnier, T. Giletti, E. Klein and L. Roques.

• Spectral synthesis in group algebras.

Eberhard Kaniuth, University of Paderborn

This is a survey talk on results in the ideal theory of Fourier and L1-algebras

of locally compact groups.

• A Poisson process model for monitoring and surveillancedata from wildlife diseases.

Volkmar Liebscher, University of Greifswald

The analysis of epidemiological field data from monitoring and surveillance

systems (MOSSs) in wild animals is of great importance to evaluate existing

systems and to specify guidelines for future measurements. Our main goal

is to implement an evaluation model for these systems.Our new approach

is based on inhomogeneous Poisson processes which describe the number of

individuals with specific epidemiological properties. For an epidemic sce-

nario, we chose an underlying SIR model which drives the intensities of the

observed process. A sampling rate has been defined which describes the

specifics of data collection for MOSSs and also takes diagnostic procedures

into account. The implementation and evaluation of the combined model

by simulation studies demonstrates its ability to validly estimate epidemic

parameters via maximum likelihood. Thus, it can help to evaluate exist-

ing disease control systems, too. The model has been tested on data from

a classical swine fever outbreak in wild boar (Rhineland-Palatinate 1999-

2002). Several extensions of the model will be discussed, too.

• Quadrature formulas on data defined manifolds.

Hrushikesh N. Mhaskar, California Institute of Technology, andClaremont Graduate University, Claremont.

A very classical problem in approximation theory is to approximate a func-

tion given its values at a finite number of points. Unlike in classical approx-

imation theory, several modern applications require that the locations at

which the target function is sampled cannot be prescribed in any structured

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manner (known as the scattered data problem). We describe the construc-

tion and properties of quadrature formulas based on scattered data that al-

lows us to study approximation using integral operators and their discretized

analogues in a unified manner. One novelty in this work is that the domain

of the target function is not known in advance, although we assume that it

is some unknown sub–manifold of a Euclidean space. It is shown in partic-

ular that the most critical aspect of these formulas is the Marcinkiewicz–

Zygmund inequalities, rather than positivity. Several theoretical results and

a few numerical examples will be presented in this direction.

• Polynomial Bases for Spaces of Continuous Functions.

Jurgen Prestin, University of Lubeck

In this talk we discuss orthogonal polynomial bases for the spaces C2π orC[−1, 1]. Of special interest are bases {pn} where pn is a polynomial of smalldegree. This problem has a long history. One milestone was achieved byR. Lorentz and A. A. Sahakian who constructed such an orthogonal basisconsisting of trigonometric polynomials of optimal degree in 1994.

Here, we consider finite-dimensional nested spaces of trigonometric polyno-mials constructed from de la Vallee Poussin means of the Dirichlet kernel.Following an approach of A. A. Privalov we investigate the correspondingMultiresolution Analysis. The scaling functions and wavelets are given ex-plicitly as trigonometric fundamental interpolants and decomposition andreconstruction algorithms can be described in simple matrix notation. Thecirculant structure of all relevant matrices allows the use of Fast-Fourier-Transform techniques for the actual implementation. Thus, we achieve al-most optimal complexity compared to other wavelet approaches derived fromimplicit two-scale relations, while dealing with a fully computable trigono-metric multiresolution analysis with explicit algebraic formulas.

Furthermore, we describe corresponding wavelet packets which yield refinedfrequency localization properties and can be used for the direct constructionof certain orthogonal polynomial bases of C2π with optimal degree. Forthe corresponding partial sum operators we obtain estimates with expliciteconstants.

The special structure of the underlying de la Vallee Poussin means allows

to transform most of these results into the algebraic case. In particular, we

obtain algebraic polynomial wavelet bases for C[−1, 1] of optimal degree.

Here the orthogonality is with respect to the Jacobi weight. The talk gives

an overview of joint work with K. Selig, R. Girgensohn and J. Schnieder.

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• Interpolation via weighted l1-minimization.

Holger Rauhut, RWTH Aachen

Functions of interest are often smooth and sparse in some sense, and both

priors should be taken into account when interpolating sampled data. Clas-

sical linear interpolation are effective under strong regularity assumptions,

but cannot incorporate nonlinear sparsity structure. At the same time, non-

linear methods such as l1-minimization can reconstruct sparse functions from

very few samples, but do not necessarily encourage smoothness. It turns out

that weighted l1-minimization effectively merges the two approaches, pro-

moting both sparsity and smoothness in reconstruction. In this talk, I will

present theoretical results on reconstruction error estimates for weighted l1-

minimization which build on concepts from compressive sensing. The theory

is underlined by numerical experiments.

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5 Scientific Program

Thursday, September 19

Time Speaker

9:00-9:15 Fabian Theis (Helmholtz Center)

Welcome address

9:15-9:30 Gero Friesecke (Dean Faculty of Mathematics TUM)

Welcome address

9:30-10:30 Eberhard Kaniuth

Spectral synthesis in group algebras10:30-11:00 Coffee

11:00-12:00 Hrushikesh N. Mhaskar

Quadrature formulas on data definied manifolds12:00- Lunch

-14:00

14:00-15:00 Jurgen Prestin

Polynomial bases for spaces of continuous functions

15:00-16:00 Hermann Eberl

Cross-diffusion in biofilms16:00-16:30 Coffee

16:30-17:30 Volkmar Liebscher

A Poisson process model for monitoring and surveillance data fromwildlife diseases

17:30-18:30 Francois Hamel

Inside structure of range-expanding populations19:15 Dinner (invited guests)

Mensa, Helmholtz Campus

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Friday, September 20

Time Speaker

9:30-10:30 Hans Georg Feichtinger

Ideas towards postmodern analysis

10:30-11:30 Karlheinz Grochenig

Wiener’s lemma in Banach algebras and norm-controlled inversion11:30- Lunch

-13:30

13:30-14:30 Holger Rauhut

Interpolation via l1-minimization

14:30-15:00 Coffee

15:00-16:00 Hartmut Fuhr

Wavelet coorbit theory in higher dimensions

16:00-17:00 Brigitte Forster-Heinlein

Phase in signals and images

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