Workshop Mathematical Analysis and Applications In occassion of the retirement of Prof. Dr. Rupert Lasser

Neuherberg, Germany, September 19 – 20, 2013

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Organizers: Wolfgang zu Castell Messoud Efendiyev Frank Filbir

Acknowledgement: The organizers would like to thank the Helmholtz Zentrum M¨ unchen and the Institute of Computational Biology for financial support. We are especially indebted to Sandra Mayer for her valuable help during the organization of the meeting.

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Contents 1 Aim

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2 Workshop Location

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3 List of Speakers

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4 Abstracts of Talks

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5 Scientific Program

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Aim

On the occasion of the retirement of Prof. Dr. Rupert Lasser the Helmholtz Zentrum M¨ unchen will organize a workshop on Mathematical Analysis and Applications Rupert Lasser was the director of the Institute of Biomathematics and Biometry at Helmholtz Zentrum M¨ unchen and had the chair for Mathematics in Medicine and Life Sciences at Technische Universit¨at M¨ unchen 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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Workshop Location

The workshop takes place at the campus of the Helmholtz Zentrum M¨ unchen.

Address: Helmholtz Zentrum M¨ unchen Ingolst¨adter Landstraße 1 D-85764 Neuherberg, Germany

Location: ”Großer H¨orsaal”, building 33, room 106.

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

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Campus Map

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List of Speakers

Hermann Eberl Department of Mathematics and Statistics, University of Guelph, Ontario, Canada Hans Georg Feichtinger Institute of Mathematics, University of Vienna, Austria Brigitte Forster-Heinlein Institute of Applied Mathematics, University of Passau, Germany Hartmut F¨ uhr Chair of Mathematics A, RWTH Aachen, Germany Karlheinz Gr¨ ochenig Institute of Mathematics, University of Vienna, Austria Fran¸cois Hamel LATP, Facult´e des Sciences et Techniques, Universit´e Aix-Marseille III, France Eberhard Kaniuth Institute of Mathematics, University of Paderborn, Germany Volkmar Liebscher Institut f¨ ur Mathematik und Informatik, Universit¨at Greifswald, Germany Hrushikesh N. Mhaskar California Institute of Mathematics, and Claremont Graduate University, Claremont, U.S.A. J¨ urgen Prestin Institute of Mathematics, University of L¨ ubeck, Germany Holger Rauhut Chair of Mathematics C, RWTH Aachen, Germany

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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 changes in our common scientific field, i.e. Harmonic Analysis to see how it has changed, and what the topics have been at the beginning and now in the late phase of our career. Since it was the time of Modern Analysis when we started I don’t know a better word for the upcoming area “Postmodern Harmonic Analysis”. After the phase of abstraction (LCA groups, distributions) and diversification (FFT, computational HA, wavelets, Gabor) and many interesting applications (e.g. in mobile communication or signal processing) it is time to think of a more integrative view-point, and bring the established theoretical body closer 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 F¨ uhr, RWTH Aachen Coorbit theory provides a functional-analytic framework for the construction and study of Banach frames arising from the action of an integrable representation. This talk is concerned with existence and basic properties of coorbit spaces associated to wavelet transforms arising from an irreducible, squareintegrable 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 actions (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 influences 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-controlled

inversion. Karlheinz Gr¨ochenig, 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 pseudodifferential 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 constructed 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. Fran¸cois Hamel, Universit´e 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 components 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 propagation 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 surveillance

data 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 scenario, 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 existing disease control systems, too. The model has been tested on data from a classical swine fever outbreak in wild boar (Rhineland-Palatinate 19992002). Several extensions of the model will be discussed, too.

• Quadrature formulas on data defined manifolds. Hrushikesh N. Mhaskar, California Institute of Technology, and Claremont Graduate University, Claremont. A very classical problem in approximation theory is to approximate a function given its values at a finite number of points. Unlike in classical approximation 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 construction and properties of quadrature formulas based on scattered data that allows 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 particular 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. J¨ urgen Prestin, University of L¨ ubeck In this talk we discuss orthogonal polynomial bases for the spaces C2π or C[−1, 1]. Of special interest are bases {pn } where pn is a polynomial of small degree. This problem has a long history. One milestone was achieved by R. Lorentz and A. A. Sahakian who constructed such an orthogonal basis consisting of trigonometric polynomials of optimal degree in 1994. Here, we consider finite-dimensional nested spaces of trigonometric polynomials constructed from de la Vall´ee Poussin means of the Dirichlet kernel. Following an approach of A. A. Privalov we investigate the corresponding Multiresolution Analysis. The scaling functions and wavelets are given explicitly as trigonometric fundamental interpolants and decomposition and reconstruction algorithms can be described in simple matrix notation. The circulant structure of all relevant matrices allows the use of Fast-FourierTransform techniques for the actual implementation. Thus, we achieve almost optimal complexity compared to other wavelet approaches derived from implicit two-scale relations, while dealing with a fully computable trigonometric multiresolution analysis with explicit algebraic formulas. Furthermore, we describe corresponding wavelet packets which yield refined frequency localization properties and can be used for the direct construction of certain orthogonal polynomial bases of C2π with optimal degree. For the corresponding partial sum operators we obtain estimates with explicite constants. The special structure of the underlying de la Vall´ee 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. Classical linear interpolation are effective under strong regularity assumptions, but cannot incorporate nonlinear sparsity structure. At the same time, nonlinear 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, promoting both sparsity and smoothness in reconstruction. In this talk, I will present theoretical results on reconstruction error estimates for weighted l1minimization which build on concepts from compressive sensing. The theory is underlined by numerical experiments.

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Scientific Program Thursday, September 19 Time

Speaker

9:00-9:15

Fabian Theis (Helmholtz Center) Welcome address Gero Friesecke (Dean Faculty of Mathematics TUM) Welcome address Eberhard Kaniuth Spectral synthesis in group algebras Coffee Hrushikesh N. Mhaskar Quadrature formulas on data definied manifolds Lunch

9:15-9:30 9:30-10:30 10:30-11:00 11:00-12:00 12:00-14:00 14:00-15:00

J¨urgen Prestin Polynomial bases for spaces of continuous functions 15:00-16:00 Hermann Eberl Cross-diffusion in biofilms 16:00-16:30 Coffee 16:30-17:30 Volkmar Liebscher A Poisson process model for monitoring and surveillance data from wildlife diseases 17:30-18:30 Francois Hamel Inside structure of range-expanding populations 19:15 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 Gr¨ochenig Wiener’s lemma in Banach algebras and norm-controlled inversion 11:30Lunch -13:30 13:30-14:30 Holger Rauhut Interpolation via l1 -minimization 14:30-15:00 Coffee 15:00-16:00 16:00-17:00

Hartmut F¨uhr Wavelet coorbit theory in higher dimensions Brigitte Forster-Heinlein Phase in signals and images

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