- ABSTRACTS -Talks/Posters

Conference on Time-FrequencyStrobl, Austria

June 15–20, 2009

CONTENTS 2

Contents1 Invited Talks 4

1.1 Candes, Emmanuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Fornasier, Massimo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Heil, Chris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Seip, Kristian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Strohmer, Thomas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Talks 72.1 Abreu, Luis Daniel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Antoine, Jean-Pierre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Balan, Radu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Balazs, Peter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Bechler, Pawel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Bishop, Shannon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 BUI, Huy-Qui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 Choudur, Lakshminarayan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.9 Christensen, Ole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.10 Czaja, Wojciech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.11 Dahlke, Stephan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.12 De Gosson, Maurice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.13 Dehghan, Mohammad Ali . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.14 Dekel, Shai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.15 Didenko, Victor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.16 Doerfler, Monika . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.17 Duits, Remco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.18 Ehler, Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.19 Englis, Miroslav . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.20 Feichtinger, Hans G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.21 Ferreira, Milton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.22 Filbir, Frank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.23 Frank, Michael . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.24 Gandy, Silvia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.25 Geller, Daryl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.26 Graef, Manuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.27 Groechenig, Karlheinz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.28 Grohs, Philipp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.29 Hansen, Markus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.30 Hellekalek, Peter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.31 Hogan, Jeff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.32 Jain, Pankaj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.33 Jang, Sumi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.34 Johansson, Karoline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.35 Jokar, Sadegh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.36 Kempka, Henning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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2.37 Kunis, Stefan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.38 Kutyniok, Gitta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.39 Kyriazis, George . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.40 Laugesen, Richard S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.41 Levitina, Tatiana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.42 Luef, Franz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.43 MacArthur, Josh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.44 Madych, W. R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.45 Makrakis, George . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.46 Narimani, Qasem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.47 Nicola, Fabio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.48 Nitzan-Hahamov, Shahaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.49 Pesenson, Isaac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.50 Petrushev, Pencho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.51 Pfander, Gotz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.52 Powell, Alex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.53 Rakhimov, Abdumalik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.54 Rauhut, Holger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.55 Sampath, Sivananthan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.56 Song, Myung-Sin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.57 Stoeva, Diana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.58 Toft, Joachim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.59 Ullrich, Tino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.60 Vasilyev, Vladimir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.61 Vourdas, Apostolos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.62 Vybiral, Jan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.63 Weisz, Ferenc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.64 Wojdyłło, Piotr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.65 Zimmermann, Georg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Posters 363.1 Carrizo, Ivana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Gole, Ioan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Heineken, Sigrid Bettina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 Maksimovic, Srdan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.5 Oztop, Serap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.6 Rieckh, Georg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.7 Romero, Jose Luis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1 Invited Talks

1.1 Emmanuel CandesCalifornia Institute of Technology, UNITED STATES

Exact Matrix Completion via Convex OptimizationThis talk considers a problem of considerable practical interest: the recovery of a data matrixfrom a sampling of its entries. In partially filled out surveys, for instance, we would like toinfer the many missing entries. In the area of recommender systems, users submit ratingson a subset of entries in a database, and the vendor provides recommendations based on theuser’s preferences. Because users only rate a few items, we would like to infer their preferencefor unrated items (this is the famous Netflix problem). Formally, suppose that we observe mentries selected uniformly at random from a matrix. Can we complete the matrix and recoverthe entries that we have not seen?

We show that perhaps surprisingly, one can recover low-rank matrices exactly from whatappear to be highly incomplete sets of sampled entries; that is, from a minimally sampled set ofentries. Further, perfect recovery is possible by solving a simple convex optimization program,namely, a convenient semidefinite program. A surprise is that our methods are optimal andsucceed as soon as recovery is possible by any method whatsoever, no matter how intractable;this result hinges on powerful techniques in probability theory. Time permitting, we will alsopresent a very efficient algorithm based on iterative singular value thresholding, which cancomplete matrices with about a billion entries in a matter of minutes on a personal computer.

1.2 Massimo FornasierJohann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian

Academy of Sciences, AUSTRIA

Compressive algorithms. Multilevel preconditioning and conver-gence rates.

Gradient iterations intertwined with adapted thresholding operations - compressive algorithms -have been recently investigated for addressing inverse problems whose solutions are character-ized by a few significant degrees of freedom. We retrace some of the history of these algorithmsand known results, and also address a variety of improved methods, such as subspace correctionmethods, or adaptations to new situations, such as free-discontinuity problems.

While the convergence of these algorithms is quite clarified, convergence rates and com-plexity are known only in special situations. In this talk we would like to focus on the com-plexity of compressive algorithms when addressing certain infinite dimensional problems. It isknown that they may perform ”arbitrarily bad” when applied for the regularized inversion ofcompact operators. Indeed for such operators the (infinite) matrix representation with respectto a ”good basis”, in the sense that it quasi-diagonalizes the operator, turns out to be diagonaldominant with fast decaying diagonal entries. The rate of convergence of the algorithms is

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related to the ”local conditioning” of such a matrix, i.e., how well-conditioned is any relativelysmall group of columns. This is the case, for instance, when we deal with potential operators,such as in magnetic tomography, and matrix representations with respect to multiscale bases orwavelets. We discuss how to precondition these problems in order to obtain a uniform conditionnumber of the resulting matrices over any small group of columns. In particular, we will showhow block-diagonal preconditioning will produce infinite matrices with a ”Restricted IsometryProperty (RIP)”, as the one introduced for finite dimensional situations in compressed sensingproblems. We will use this property in order to show how adaptive numerical iterations can beperformed guaranteeing a controlled linear convergence of these algorithms.

1.3 Chris HeilGeorgia Tech., USA, UNITED STATES

Linear Independence of Gabor Systems and Other Open ProblemsGabor frames have been widely studied and have many applications, yet one of the ”simplest”questions remains unanswered to this day: Are all Gabor systems finitely linearly independent?We will discuss some of the history and known results on this question, and also survey a varietyof other open problems in frame theory and time-frequency analysis.

see also:http://univie.ac.at/nuhag-php/dateien/talks/1325_strobl09.pdf

1.4 Kristian SeipNorwegian University of Science and Technology (NTNU), NORWAY

The Feichtinger conjecture in model subspaces of H2

The Paley-Wiener space is the prime example of a so-called model subspace of the Hardy spaceH2. The interesting question of whether the Feichtinger conjecture holds true for sequences ofnormalized reproducing kernels in every model subspace is probably not much easier than thegeneral version of the conjecture. I will give some background for this problem and presentwhat is known about it.

1.5 Thomas StrohmerUniversity of California Davis, UNITED STATES

From Helmholtz to Heisenberg: Sparse Remote SensingWe consider the problem of detecting targets via remote sensing. This imaging problem is typ-ically plagued by nonuniqueness and instability and hence mathematically challenging. Tra-ditional methods such as matched field processing are rather limited in the number of targets

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1 INVITED TALKS 6

that they can reliably recover at high resolution. By utilizing sparsity and tools from com-pressed sensing I will present methods that significantly improve upon existing radar imagingtechniques. I will derive fundamental performance and resolution limits for compressed radarimaging with respect to the number of sensors and resolvable targets. These theoretical resultsdemonstrate the advantages as well as limitations of compressed remote sensing. Numericalsimulations confirm the theoretical analysis. This is joint work with Albert Fannjiang, MikeYan, and Matt Herman.

2 Talks

2.1 Luis Daniel AbreuUniversity of Coimbra, FCTUC, Department of Mathematics, PORTUGAL

Sampling and interpolation in Bargmann-Fock spaces of polyanalyticfunctions

Vector-valued Gabor frames can be used to transmit several signals encoded into a single one(Multiplexing). An important breakthrough in this research topic has been obtained recentlyby Grchenig and Lyubarskii, by giving a characterization of all the lattices that generate super-frames with Hermite windows (Math. Ann. 2009). In this talk we will show that Grchenig-Lyubarskii result is a part of a ”sampling and interpolation density theorem” for Fock spacesof polyanalytic functions, in the lattice case. We will prove this theorem in its full general-ity, by using the vectorial version of Janssen-Daubechies-Landau-Ron-Shen duality principleto reduce the problem to a result of Brekke and Seip (Math. Scand. 1993). The theorem dis-plays a ”Nyquist rate” which increases with n, the degree of polyanaliticity of the space. Wewill give some evidence of why polyanalytic functions are the right complex analysis tools totransmit vector valued data, in situations where analytic functions transmit single signals. Forinstance, the same thing happens in Wavelet analysis. We introduce polyanalytic versions of theBargmann and the Bergman transform. This ideas are connected to the theory of superframesdeveloped by Balan, Han and Larson in a series of papers. Moreover, the polyanalytic Fockspaces are eigenspaces of the Landau equation with magnetic field and can be used as modelsfor displaced Fock states (the states based on the excited states of the harmonic oscillator incontrast to the ground-state based Schrdinger coherent states). Polyanalytic functions were in-troduced 101 years ago by the Russian mathematician G. V. Kolossov in his work on the theoryof elasticity (C. R. Acad. Sci. Paris 1908) and they were studied intensively as pure mathemat-ical objects in the 70/80s by the Russian school lead by M. B. Balk. However, the connectionwith time-frequency analysis seems to have been hitherto unnoticed. To have an intuitive graspof the results one can think that the vector formed with the first n Hermite functions plays, invectorial Gabor analysis, the same structural role as the Gaussian in scalar Gabor analysis.

2.2 Jean-Pierre AntoineUniversite catholique de Louvain, BELGIUM

Generalized frames: weighted, controlled, unbounded(Joint work with Peter Balazs and Anna Grybos)

Frames, especially tight frames, are a convenient substitute to (bi-)orthogonal bases inwavelet and Gabor analysis. In particular, discretizing a continuous wavelet transform leadsin general to a frame. However, it turns out that the resulting expansions often converge ratherslowly, for instance in the case of wavelets on the sphere. In order to improve the situation,

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two different generalizations of the concept of frames have been introduced, weighted andcontrolled frames.

A sequence Ψ = (ψn)n∈Γ of elements in a Hilbert space H is a weighted frame of H, withpositive weights (wn)n∈Γ, if there exist constants m > 0 and M <∞ such that

m‖f‖2 6∑n∈Γ

wn |〈f, ψn〉|2 6 M‖f‖2, ∀ f ∈ H.

A sequence Ψ = (ψn)n∈Γ ∈ H is a frame controlled by the positive operator C, boundedwith bounded inverse, if there exist constants m > 0 and M <∞ such that

m‖f‖2 6∑n

〈ψn, f〉 〈f, Cψn〉 6 M‖f‖2, ∀ f ∈ H.

In the first part of the talk, we will examine some properties of these two objects and, in partic-ular, their mutual relationship. We also give some numerical results concerning the choice ofweights (wn) [1].

In a second part, we will discuss another generalization of standard frames, namely, un-bounded frames. These are simply frames for which the lower frame bound m = 0, in otherwords, frames whose frame operator has an unbounded inverse. This notion has been intro-duced, for continuous frames, in [2, Sec.7.3]. We will particularize it to the discrete case andexamine the corresponding synthesis formula. The interesting point is that one has to work ina Gel’fand triplet, consisting of three reproducing kernel Hilbert spaces.

References[1] P. Balazs, J-P. Antoine, and A. Grybos, Weighted and controlled frames: Mutual relationship and

first numerical properties, Int. J. Wavelets, Multires. and Inform. Proc. 2009 (to appear)

[2] S.T. Ali, J-P. Antoine, and J-P. Gazeau, Coherent States, Wavelets and Their Generalizations,Springer, New York et al., 2000

see also:http://univie.ac.at/nuhag-php/dateien/talks/1268_antoine-abs_strobl2009.pdf

2.3 Radu BalanDepartment of Mathematics and CSCAMM, University of Maryland, UNITED STATES

Redundancy for Localized FramesRedundancy is the qualitative property which makes Hilbert space frames so useful in practice.However, developing a meaningful quantitative notion of redundancy for infinite frames hasproven elusive. Though quantitative candidates for redundancy exist, the main open problemis whether a frame with redundancy greater than one contains a subframe with redundancyarbitrarily close to one. We will answer this question in the affirmative for `1-localized frames.We then specialize our results to Gabor multi-frames with generators in M1(Rd), and Gabor

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molecules with envelopes inW (C, l1). As a main tool in this work, we show there is a universalfunction g(x) so that for every ε > 0, every Parseval frame {fi}Mi=1 for an N -dimensionalHilbert space HN has a subset of fewer than (1 + ε)N elements which is a frame for HN withlower frame bound g(ε/(2M

N− 1)). This is joint work with Pete Casazza and Zeph Landau.

2.4 Peter BalazsAcoustics Research Institute, AUSTRIA

Frames and Hilbert-Schmidt OperatorsHere we deal with the theory of Hilbert-Schmidt operators HS, when the usual choice oforthonormal basis, on the associated Hilbert spaces, is replaced by frames. More precisely, weprovide a necessary and sufficient condition for an operator to be Hilbert-Schmidt, based on itsaction on the elements of a frame (i.e., an operator T isHS if and only if the sum of the squarednorms of T applied on the elements of the frame is finite.) Also, we construct Bessel sequences,frames and Riesz bases of HS operators using tensor products of the same sequences in theassociated Hilbert spaces. We state how the HS inner product of an arbitrary operator and arank one operator can be calculated in an efficient way; and we use this result to provide anumerically efficient algorithm to find the best approximation, in the Hilbert Schmidt sense,of an arbitrary matrix, by a so-called frame multiplier (i.e., an operator which act diagonallyon the frame analysis coefficients.) Finally, we give some simple examples using Gabor andwavelet frames, introducing in this way wavelet multipliers.

2.5 Pawel BechlerInstitute of Applied Mathematics and Mechanics Faculty of Mathematics, Informatics and Me-

chanics, Warsaw University, POLAND

Orthogonal matching pursuit and dictionaries with restricted isome-try property

We study the Orthogonal Matching Pursuit with regard to a dictionary with restricted isometryproperty in a Hilbert space. We obtain upper estimates for the error of OMP in terms of theerror of the best n-term approximation (Lebesgue-type inequalities). This is a development ofrecent results obtained by D.L. Donoho, M. Elad and V.N. Temlyakov. This is joint work withP. Wojtaszczyk.

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2.6 Shannon BishopGeorgia Institute of Technology, UNITED STATES

Mixed Modulation Spaces and Pseudodifferential OperatorsWe will use frame techniques to characterize the Schatten class properties of integral operators.If the coefficients {〈k,Φm,n〉} of certain frame expansions of the kernel k of an integral operatorare in `2,p, then the operator is Schatten p-class. As a corollary, we conclude that if the kernel orKohn-Nirenberg symbol of a pseudodifferential operator lies in a particular mixed modulationspace, then the operator is Schatten p-class. Our corollary improves existing Schatten classresults for pseudodifferential operators and the corollary is sharp in the sense that larger mixedmodulation spaces yield operators that are not Schatten class.

2.7 Huy-Qui BUIUniversity of Canterbury, NEW ZEALAND

New and old Besov spacesIn this talk I will introduce a new class of Besov spaces associated with an operator L. Here Lis a densely defined operator on L2(X), and (X, d, µ) is a quasi-metric space equipped with anon-negative Borel measure µ, in which the measures of open balls have polynomial growth. Arich theory can be developed when L generates a holomorphic semigroup and has heat kernelbounds.

The main aim of the talk is to show that, when the heat kernels also satisfy the Holdercontinuity and cancellation properties, then the new Besov spaces are equivalent to the classicalones.

This is joint work with Xuan Thinh Duong (Macquarie University, Australia) and Lixin Yan(Zhongshan University, China).

see also:http://univie.ac.at/nuhag-php/dateien/talks/1244_nhgslides.pdf

2.8 Lakshminarayan ChoudurHewlett-Packard Laboratories, UNITED STATES

Adaptive Wavelet Filtering in Database ApplicationsIt is well known in the wavelets literature that only a few wavelet coefficients contribute tothe signal while a preponderance of coefficients are below the noise level in the system. Thehard and soft thresholding procedures provide a methodology for coefficient selection for goodcompression. In modern database applications, time and main memory limitations demandhigh compression ratios and good reconstruction accuracy. In one application in databases, asynopsis of a data distribution is stored as a histogram which the query optimizer uses at com-pile time to determine the best access path to produce a desirable query plan. More recentlyadaptive spatial procedures are becoming increasingly popular to analyze and synthesize data

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distributions in databases as alternatives to histograms. In this connection, function approx-imation by wavelets and compression by thresholding are proposed as an alternative. Afterselecting a wavelet basis for function approximation, we introduce a method known as energybased thresholding to obtain compression. We posit that by invoking the Bessels inequality;, asimple plot of the cumulative energy versus number of coefficients, provides the mechanism toselect the number of coefficients needed to achieve various levels of reconstruction accuracy.In other words, coefficients can be chosen to accommodate accuracy and storage requirements.In other situations, the user wants to determine the number of coefficients needed to meet a pre-specified level of accuracy. This is helpful to estimate trade-offs between storage space neededand accuracy of reconstruction. This is the converse of the problem we looked at previously,where we picked the number of coefficients that captured a certain amount of information inreconstructing original data within desired accuracy levels. In essence, the energy informationis used to provide flexibility to achieve desirable trade offs between compression and errors inreconstruction of the original data towards accommodating memory constraints.

The attractiveness of energy based thresholding is the ability to choose coefficients flexiblythat it is absent in the popular hard and soft thresholding methods.

Furthermore, we notice that Hard and soft thresholding admit coefficients which in mag-nitude are smaller than the noise level and contribute little to reconstructing the signal. Whileour proposal overcomes this problem to some degree, we alternatively propose a methodologyto shrink these coefficients to zero. Detailed studies and experiments using a variety of func-tions often arising in signal processing, spectroscopy and image processing (Blocks, Bumps,Heavy-Sine, Doppler, Mishmash, Quadchirp) indicate that our methods yield significant com-pression with high reconstruction accuracy. The proposed methods explicitly take into accountthe relationship between the energy encapsulated in the data and the wavelet transform.

2.9 Ole ChristensenTechnical University of Denmark, DENMARK

Pairs of dual Gabor framesConsider a bounded function g supported on [−1, 1] and a modulation parameter b ∈]1/2, 1[for which the Gabor system {EmbTng} is a frame. We show that such a frame always has acompactly supported dual window. More precisely, we show that if b < N

N+1for some N it

is possible to find a dual window supported on [−N,N ]. Under the additional assumption thatg is continuous and only has a finite number of zeros on ] − 1, 1[, we characterize the frameproperty of {EmbTng}. As a consequence we obtain easily verifiable criteria for a function g togenerate a Gabor frame with a dual window having compact support of prescribed size.

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2.10 Wojciech CzajaUniversity of Maryland, POLAND

Kaczmarz algorithms and framesIn 1937 Stefan Kaczmarz introduced an iterative method for solving linear systems of equa-tions. Recently, Kwapien and Mycielski, and Szwarc and Haller characterized convergentKaczmarz algorithms in infinite-dimensional Hilbert spaces by means of tight frames of ef-fective sequences. The purpose of this talk is to extend some of their results by introducing andcharacterizing a more general notion of convergence of the algorithm, as well as introducingframes and Riesz bases in this context.

2.11 Stephan DahlkeUniversity of Marburg, GERMANY

The continuous shearlet transform in arbitrary space dimensionsWe shall be concerned with the generalization of the continuous shearlet transform to higherdimensions. Similar to the two-dimensional case, our approach is based on translations,anisotropic dilations and specific shear matrices. We show that the associated integral transformagain originates from a square-integrable representation of a specific group, the full n-variateshearlet group. Moreover, we verify that by applying the coorbit theory, canonical scales ofsmoothness spaces and associated Banach frames can be derived. We also indicate how ourtransform can be used to characterize singularities in signals.

This is joint work with G. Steidl and G. Teschke

2.12 Maurice De GossonNuHAG, Faculty of Mathematics, University of Vienna, AUSTRIA

The symplectic camel: the tip of an iceberg?

2.13 Mohammad Ali Dehghanspeaker, IRAN, ISLAMIC REPUBLIC OF

Linear Operators that preserve some properties of FramesThe presentation centers around a bounded linear operator that preserves duality , frame oper-ator and conjugate frames in a separable Hilbert space. Some operators respect to conjugateframes will be introduced and their relations will be established.

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2.14 Shai DekelGE Healthcare and Tel-Aviv university, Israel, ISRAEL

Adaptive compressed sensing using direct sampling of dictionariesWe present a sampling process that enables to acquire and compress high resolution data, with-out fully acquiring the entire data set at its highest resolution, using significantly less measure-ments. In some cases, our approach simplifies and improves upon the existing methodologyof the new and emerging field of compressed sensing, by replacing the universal acquisitionof pseudo-random measurements with a direct and fast method of adaptive acquisition fromdictionaries. We will focus on the example where the data is an 2D image and the dictionaryis composed of wavelet and windowed Fourier. This combination is designed to address themodel of images as a blend of a cartoon with local texture patches. Whereas some compressedsensing algorithms are not computationally feasible for large data sets, our method is very fastregardless of the image size. Joint work with A. Averbuch and S. Deutsch.

2.15 Victor DidenkoUniversiti Brunei Darussalam, BRUNEI DARUSSALAM

On L2-Solutions of Refinement EquationsIn this talk we present various properties of L2-solutions of the operator equations

f −BMFaF−1f = g (1)

where a is the operator of multiplication by a matrix a ∈ Lm×m∞ (Rs), m, s ∈ N, F denotes theFourier transform, and BM : Lm2 (Rs) 7→ Lm2 (Rs) is the dilation operator

BMf(x) := f(Mx), x ∈ Rs

generated by a non-singular matrix M ∈ Rm×m. Note that the class of equations (1) containsdiscrete and continuous refinement equations widely used in wavelet analysis.

It is shown that the set of nontrivial solutions of the homogeneous equation is either emptyor contains a subset isomorphic to a space L∞(VM), where VM is a Lebesgue measurable setwith a positive Lebesgue measure. Therefore, the operator I−BMFaF−1 : Lm2 (Rs) 7→ Lm2 (Rs)is Fredholm if and only if it is invertible. Moreover, various properties of L2-solutions of bothhomogeneous and non-homogeneous forms of equation (1) are established. In particular, if thedilation M satisfies some mild conditions, then for each refinement operator I−BMFaF−1 theinclusion ker (I −BMFaF−1) ⊂ im (I −BMFaF−1) holds.

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2.16 Monika DoerflerNuHAG, Faculty of Mathematics, University of Vienna, AUSTRIA

Quilted Gabor frames - progress and challengesAll commonly used classes of frames, be it wavelet or Gabor frames, feature a resolution fol-lowing a fixed rule over the whole time-frequency or time-scale plane, respectively. In this talk,the new concept of ”quilted frames” is introduced, which gives up this uniformity and allowsfor different resolutions in assigned areas of the time-frequency plane. The primary motivationstems from the processing of audio and in particular music signals, where the trade-off betweentime- and frequency resolution has a strong impact on the results of analysis and synthesis. Wewill introduce the general concept and describe some special cases for which the frame propertycan be proved.

2.17 Remco DuitsEindhoven University of Technology, Dep. of Mathematics and Computer Science + Dep. of

Biomedical Engineering, NETHERLANDS

Left Invariant Convection and Diffusion on Gabor TransformsWe consider enhancement techniques for Gabor transforms based on left-invariant PDE’s de-fined on the reduced Heisenberg group. We define adaptive, nonlinear transport and diffusionequations on Gabor transforms, either in order to enhance the sharpness of the time-frequencyrepresentation, or for enhancement of the reconstructed signal. More specifically, we distin-guish between three types of processing on Gabor transforms. First we consider non-linearadaptive left-invariant convection (reassignment), while approximately maintaining the origi-nal signal. (This procedure is similar to the reassignment methods studied by various authors,e.g. Chassande-Mottin, Daubechies, Flandrin,Torresani etc.) Secondly we consider non-linearadaptive left-invariant diffusions on Gabor transforms for denoising and signal enhancement,beyond the well-known soft-thresholding algorithms. Thirdly we apply recent Euler-Lagrangetechniques on a contact manifold within Hr in the computation of geodesics and snakes in theGabor domain and investigate in what sense the left-invariant reassignment algorithms con-centrate to these curves in the Gabor domain. For implementation purposes, it is beneficialto transfer the algorithms from the (three-dimensional) reduced Heisenberg group to (two-dimensional) phase space. We provide numerical algorithms for the 3 approaches with practi-cal experiments and analytical examples of left-invariant reassignment on Gabor transforms ofchirp signals.

Finally, we show how the presented methods fit in a larger group theoretical framework.Here we provide explicit connections to our previously developed left-invariant non-linear evo-lution equations on the 2D-Euclidean motion group for enhancement and completion of cross-ing lines/contours in 2D-images.

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2.18 Martin EhlerNorbert Wiener Center, UNITED STATES

Minimization with sparsity constraints for demixing multispectraldata to analyze molecular processes within the human retina

Many retinal diseases are associated with the distribution of fluorescent photochemicals thataccumulate within the human retina. Hence detection and classification of these molecularphotoproducts are important for the evaluation of early drug interventions.

A multispectral datacube is acquired by imaging patients’ retina at different wavelengths.Each pixel is then a vector and represents the spectral mixture of pure substances. One of themajor challenges is to determine these substances and then demixe each pixel.

One commonly assumes that there are only few pure substances. Each pixel vector thenhas a sparse representation. Our proposed algorithm solves a series of minimization problemswith sparsity constraints to recover the pure substances. This algorithm is applied to images ofNational Eye Institute study patients with retinal pathology, but could be usefully applied to awide range of demixing problems.

2.19 Miroslav EnglisMathematics Institute AS CR, Prague, CZECH REPUBLIC

Toeplitz quantization on real symmetric domainsBased on group-theoretic considerations, we study the analogue of the Toeplitz (anti-Wick)operator calculus in the setting of real symmetric domains, including an analogue of the star-product familiar from the Berezin-Toeplitz quantization. The analogue turns out to be a certaininvariant operator, which one might call star restriction, from functions on the complexificationof the domain into functions on the domain itself. In particular, we establish the usual (i.e. semi-classical) asymptotic expansion of this star restriction, and describe real-variable analogues ofseveral other results.

2.20 Hans G. FeichtingerNuHAG, Faculty of Mathematics, University Vienna, AUSTRIA

Gelfand Triples, the kernel theorem and Diracs symbolic calculusWe will indicate how the theory of Banach Gelfand Triples (a variant of the so-called riggedHilbert spaces used in quantum mechanics), based on the Segal algebar S0(Rd) can be used togive a meaning to the expressions arising in Dirac’s calculus, using the bra-kets. The Fouriertransform is the perfect example, showing that one cannot stay within the Hilbert space set-ting of L2(Rd), because its building blocks -S the pure frequencies -are not square integrable.Moreover they form a continuously parametrized family (however not a continuous frame). Thekernel theorem as well as the composition law - described at the kernel level - gives a way to

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reinterpret some of the results given by Dirac as statements about the composition of (unitary)Banach Gelfand Triple isomorphisms. In this way also the notational connection between theKronecker δ-symbol and the Dirac symbols can be pointed out in a natural way. In fact, fromour point of view it is a way to describe the analogue of unitary matrices U over Cn, whichhave the property that U ′ ∗ U = Idn = U ∗ U ′ , where U ′ describes the adjoint (transpose +conjugate) matrix. Special emphasis is put on the relevance of w∗-convergence within the dualspace of S0(Rd).

2.21 Milton FerreiraUniversidade de Aveiro, PORTUGAL

The inversion of the Radon transform on SO(3) by Gabor frameexpansions

The inversion of the Radon transform on SO(3) by Gabor frame expansionsJoint work with Gerd Teschke, Uwe Kahler and Paula Cerejeiras.This talk is concerned with the problem of the inversion of the one-dimensional Radon

transform on the rotation group SO(3) and its application to X-Ray tomography with poly-crystalline materials. The proposed approach is composed by Gabor frames constructed fromthe work of B. Torrsani about local Fourier analysis on spheres and through the coorbit theoryon homogeneous spaces. For the numerical solution of the problem we use variational princi-ples for sparse reconstructions that yield iterative approximation of the solution of the inverseproblem.

2.22 Frank FilbirInstitute of Biomathematics and Biometry, Helmholtz Center Munich, GERMANY

Kernel based approximation on manifoldsLet {φj} be an orthonormal system on a quasi-metric measure space X, {`j} be a nondecreasingsequence of numbers with limj→∞ `j = ∞. A diffusion polynomial of degree L is an elementof the span of {φk : `l ≤ L}. We study approximation processes of the form

σLf(x) =∞∑j=0

H(`jL

)〈f, φj〉φj(x),

where H is a suitable function. First, we address the problem under which conditions we canexpect convergence, i.e. ‖σLf − f‖p → 0, L → ∞. Secondly, we will consider the relationbetween the localization of the kernel

ΦL(x, y) =∞∑j=0

H(`jL

)φj(x)φj(y),

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the wave kernel W (t, f1, f2) =∑∞

j=0 cos(`jt)〈f1, φj〉 〈f2, φj〉, and a generalized heat kernelKt(x, y) =

∑∞j=0 exp(−`2

j t)φj(x)φj(y) respectively.Finally we will address the problem of deriving Marcinkiewicz-Zygmund type inequalities

for scattered data on a Riemannian manifold.

This talk is based on joint work with Hrushikesh N. Mhaskar, Department of Mathematics,California State University, U.S.A.

References[1] F. Filbir, H. N. Mhaskar, A quadrature formula for diffusion polynomials corresponding to

a generalized heat kernel, submitted 2009.

[2] F. Filbir, H. N. Mhaskar, J. Prestin, On a filter for exponentially localized kernels based onJacobi polynomials, to appear in J. Approx. Theory.

[3] M. Maggioni, H. N. Mhaskar, Diffusion polynomial frames on metric measure spaces,Appl. Comput. Harmon. Anal. 24: 329–353, 2008.

[4] C. K. Chui, D. L. Donoho, Special Issue: Diffusion Maps, Appl. Comput. Harmon. Anal.21 : 2006

2.23 Michael FrankHochschule fur Technik, Wirtschaft und Kultur, Leipzig, GERMANY

Orthogonality-preserving and conformal mappings on Hilbert C*-modules

The class of bounded linear maps between two Hilbert spaces mapping Parseval frames to Par-seval frames or to tight frames, respectively, are the coisometries and the linear conformal map-pings. The latter set coincides with the set of all coisometries multiplied by non-zero constants.The more, on Hilbert spaces every orthogonality preserving linear map is conformal automati-cally. We consider the more general situation of standard frames on Hilbert C∗-modules fromthe opposite side. A classification of orthogonality-preserving bounded module maps is given.The geometric background can be described in terms of certain modular hyperplanes. The setof conformal bounded module maps is shown to be strictly smaller in case of a non-trivial cen-ter of the C∗-algebra of coefficients. The situation for the characteristics of mapped Parseval ortight frames is demonstrated. In particular, the results are of interest for parametrized familiesof frames in continuous fields of Hilbert spaces.

see also:http://univie.ac.at/nuhag-php/dateien/talks/1260_MFrank_Strobl09.pdf

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2.24 Silvia GandyTokyo Institute of Technology, JAPAN

A study of low-rank matrix recovery from independent linear mea-surements of the column vectors

The problem of finding a matrix of minimal rank that fits a set of linear constraints is a NP-hardproblem. However, it was shown that if the linear transformation defining the constraints fulfillsa certain restricted isometry property, one can retrieve the minimum rank solution by solving aconvex optimization problem. As cost function of the optimization problem the nuclear norm isbeing used. Previous results can be grouped into two different settings. The first setting assumesthe absence of a special structure of the linear transformation describing the constraints. Thesecond setting describes the special case of matrix completion (where entries of the matrixare sampled). We studied a different setting, enforcing a structural constraint on the linearconstraint set. The restriction was introduced that the measurements act independently on theindividual columns of the matrix which we wish to recover.

The optimization problem can be formulated as follows. Let As ∈ Rd×n1 be real matrices,for s ∈ {1, . . . , n2}. Determine a matrix X that minimizes

minimize ‖X‖∗subject to Asxs = bs, ∀s ∈ {1, . . . , n2},

where X = [x1, . . . , xn2 ] ∈ Rn1×n2 and xs are the column vectors of X . The matrices As shallbe so-called nearly isometric random matrices.

We show that this setting — although the only information connecting the column vectors isthe low-rank requirement — is powerful enough to recover low-rank matrices and give a boundon the number of constraints necessary. Also, we present the results of numerical experimentsthat illustrate its performance.

2.25 Daryl GellerDepartment of Mathematics, Stony Brook University, NY, UNITED STATES

Spin Wavelets on the Sphere for CMB Polarization Data AnalysisThe cosmic background radiation (CMB), which was emitted only 400,000 years after the BigBang, has attracted an enormous amount of attention. Physics was very simple then, so ananalysis of the data can be used, to confirm or eliminate various physical theories which havebeen proposed concerning the universe, and to estimate numerous physical parameters, such asthe amount of dark matter in the universe. In order to do this, however, one must have a way ofextracting statistically valid estimators from the data. Since CMB cannot be observed in a largeportion of the sky, owing to interference from the Milky Way, it is not possible to use sphericalharmonic coefficients as such estimators. Spherical wavelet coefficients can be used effectivelyinstead. Because wavelets are well-localized, one can to a great extent avoid the unobservedregion.

CMB has both temperature and polarization. Almost all work to date has been on temper-ature, for which precise data is available. We study polarization, for which precise data should

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be available soon. Polarization data is not an ordinary function on the sphere, but rather a(random) section of a line bundle. Starting with the spin spherical harmonics of Newman andPenrose, we construct spin wavelets on the sphere, which are appropriate for analyzing sectionsof this line bundle. We show that spin wavelet coefficients have the required statistical prop-erties, under reasonable assumptions. (The needlets of Narcowich, Petrushev and Ward wereearlier used by Baldi, Kerkyacharian, Marinucci and Picard for similar purposes, in studyingCMB temperature.) One hopes to use CMB polarization to provide the first direct evidence ofgravitaional waves.

This is joint work, contained in three articles, one jointly with Marinucci, another jointlywith Mayeli, and another (in Phys Rev D) jointly with Hansen, Marinucci, Kerkyacharian, andPicard.

2.26 Manuel GraefChemnitz University of Technology, GERMANY

Sampling sets and quadrature formulae on the rotation groupIn this talk we construct sampling sets over the rotation group SO(3). The proposed construc-tion is based on a parameterization, which reflects the product nature S2xS1 of SO(3) verywell, and leads to a spherical Pythagorean-like formula in the parameter domain. We provethat by using uniformly distributed points on S2 and S1 we obtain uniformly sampling nodeson the rotation group SO(3). Furthermore, quadrature formulae on S2 and S1 lead to quadra-tures on SO(3), as well. For scattered data on SO(3) we give a necessary condition on themesh norm such that the sampling nodes possesses nonnegative quadrature weights. We pro-pose an algorithm for computing the quadrature weights for scattered data on SO(3) based onfast algorithms. We confirm our theoretical results with examples and numerical tests.

2.27 Karlheinz GroechenigNuHAG, Faculty of Mathematics, University of Vienna, AUSTRIA

Four years of the European Center of Time-Frequency AnalysisIn this talk I will present some highlights of the work of the European Center of Time-Frequency Analysis.

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2.28 Philipp GrohsTU Wien, AUSTRIA

Scaling functions for Shearlet MRAsIn my talk I report on ongoing work adressing the problem of making directional decomposi-tions of bivariate functions computationally efficient. The approach is to construct a Multires-olution Analysis with a scaling function that satisfies a refinement relation especially suitedto the shearlet transform. Such functions can be constructed from a certain novel subdivisionprocedure.

2.29 Markus HansenFriedrich-Schiller-Universitaet Jena, GERMANY

Best m-term Approximation and Lizorkin-Triebel SpacesWe are interested in the behaviour of the widths

σm(X, Y,B) := sup{σm(f, Y,B) :

∥∥f ∣∣X∥∥ ≤ 1}, m ∈ N ,

where X and Y are quasi-Banach spaces, cb ⊂ X is a fixed set, such that its span is densein X , and σm(f, Y, cb) is the error of the best m-term approximation with respect to cb in thequasi-norm of Y .

Here we investigateX and Y being Lizorkin-Triebel as well as Besov spaces. This includesthe case Y = Lp(Rd), which is of paramount importance. Our method allows us to treathomogeneous and inhomogeneous spaces more or less simultaneously.

We supplement known results of DeVore, Popov and Kyriazis.

2.30 Peter HellekalekDr. Peter Hellekalek Fachbereich Mathematik Universitat Salzburg, AUSTRIA

On p-adic characters in the theory of uniform distribution of se-quences mod 1

The classical theory of uniform distribution of sequences on the s-dimensional torus rests uponone –orthogonal– function system, the system of trigonometric functions.

In this talk, we will discuss a more general approach. The central qualitative and quan-titative results of the theory of uniform distribution of sequences, Weyl’s Criterion and theinequality of Erdos-Turan-Koksma, will be considered in this general setting. In particular, wewill study a function system related to the dual group Zp of the p-adic integers Zp (p a primenumber) and derive qualitative and quantitative results for the uniform distribution of sequencesmodulo one.

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2.31 Jeff HoganUniversity of Newcastle, AUSTRALIA

The Clifford-Fourier transformIn this talk we attempt to synthesize recent progress made in the mathematical and electricalengineering communities on topics in Clifford analysis and the processing of multichannel sig-nals, in particular the construction and application of Fourier transforms which arise naturallyfrom the Clifford analysis of euclidean spaces. Emphasis will be placed on the two-dimensionalsetting where the appropriate underlying Clifford algebra is the familiar set of quaternions.We’ll describe some results and problems in the construction of discrete wavelet bases forcolour images, and the difficulties encountered in constructing Clifford-Fourier kernels in di-mensions 3 and higher.

2.32 Pankaj JainUniversity of Delhi, INDIA

Grand Function Spaces and Related InequalitiesWe shall discuss grand Lebesgue and grand Lorentz spaces. Many properties of these spaceswill be discussed. A number of inequalities involving Hardy-type operator and maximal oper-ator will be studied in the framework of grand Lebesgue and grand Lorentz spaces.

2.33 Sumi JangKorea Advansed Institute of Science and Technology(KAIST), Daejeon, KOREA, REPUBLIC

OF

Multiwindow tight Gabor framesFor positive integers N and M , let gn ∈ L2(R) with supp(gn) = [0,M ], n = 1, 2, ..., N , wegive a neccessary and sufficient condition for the multiwindow Gabor systemG := {gn;k,l(t) :=gn(t−k)e2πilt}n=1,...,N ; k,l∈Z to be a tight frame by the properties of the matrix A(t)∗A(t) whereA(t) = (gn(t −m))N×M . We also show that G froms an orthonormal basis for L2(R) if andonly if N = 1 and |g1(t)| =

∑M−1m=0 χm+Em(t) a.e. where {Em}m=0,...,M−1 forms a Lebesgue

measurable partition of the unit interval [0, 1]. Our criterion provides a rich family of tightmultiwindow Gabor fames for L2(R).

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2.34 Karoline JohanssonVaxjo University, SWEDEN

Wave-front sets of Fourier Banach and modulation space typesWe will discuss some properties of wave-front sets of (weighted) Fourier Banach type. Wave-front sets have been introduced by Hrmander for classical and Sobolev types and by Pilipovic,Teofanov and Toft for Fourier Lebesgues types. We prove that some properties that are validfor the classical wave-front sets (shown by Hrmander), and for wave-front sets of weightedFourier Lebesgues types, for a broad class of pseudo-differential operators (shown by Pilipovic,Teofanov and Toft), also hold in the case of weighted Fourier Banach types. The pseudo-differential operators that we consider have smooth symbols a ∈ S(ω)

ρ .Here we consider Banach spaces B that are translation invariant BF-spaces and weights ω

which are moderated by some polynomial. Let

|f |FBΓ(ω)

= ‖fω(x, ·)χΓ(·)‖B, (2.2)

where for χΓ is the characteristic function with respect to Γ and Γ is an open cone. LetΘFB(ω)

(f) be the set of all ξ in Rd\0 such that (2.2) is finite for some Γ which contains ξ. Thenlet ΣFB(ω)(f) be the complement of ΘFB(ω)

(f) in Rd\0. Let X be an open subset of Rd\0,f be a distribution defined on X and ω0(ξ) = ω(y0, ξ) for some y0 ∈ Rd. Then we constructthe wave-front set of weighted Fourier Banach type as the set of all pairs (x0, ξ0) ∈ X ×Rd\0such that

ξ0 ∈ ΣFB(ω0)(φf) = ΣFB(ω)

(φf)

for each φ ∈ C∞0 (X) such that φ(x0) 6= 0. The wave-front set of weighted Fourier Banach typeis denoted by WFFB(ω)(f).

2.35 Sadegh JokarTU Berlin, GERMANY

Kronecker Products and Conditions for Uniqueness of the SparseSolutions to Underdetermined Systems

The Kronecker product of matrices plays a central role in mathematics and in applicationsfound in engineering and theoretical physics. Firstly we study three properties of matrices: thespark, the mutual incoherence and the restricted isometry property which have recently beenintroduced in the context of compressed sensing for matrices that are Kronecker products andshow how these properties relate to those factors. Then we consider the null space property andits relation to the restricted isometry property. Finally we give a sufficient condition for a givenmatrix to satisfy the null space property in terms of k-th mutual incoherence.

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2.36 Henning KempkaDepartment of Mathematics, University of Jena, GERMANY

Wavelet decomposition of function spaces with variable smoothnessand integrability

We introduce 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability.These spaces are defined Fourier analytical and we give a characterization by decompositionwith wavelets. These spaces cover the usual Besov and Triebel Lizorkin spaces as well asspaces of variable smoothness and integrability.

2.37 Stefan KunisFaculty of Mathematics, Chemnitz University of Technology, GERMANY

High dimensional fast Fourier transformsA straightforward discretisation of high dimensional problems often leads to an exponentialgrowth in the number of degrees of freedom. Thus, even efficient algorithms like the fastFourier transform have high computational costs. Sparse approximations allow for a severe de-crease in the number of used Fourier coefficients to represent functions of appropriate smooth-ness. Of course, an import issue is the adaption of efficient algorithms to these thinner discreti-sations such that their total complexity is within logarithmic factors still linear in the decreasedproblem size. We discuss sparse and hyperbolic cross fast Fourier transforms with arbitraryand dedicated sampling schemes.

2.38 Gitta KutyniokUniversitat Osnabruck, GERMANY

A Sparsity Approach to the Geometric Separation ProblemModern data are often composed of two (or more) geometrically distinct constituents - forinstance, pointlike and curvelike structures in astronomical imaging of galaxies. Although itseems impossible to extract those components - as there are two unknowns for every datum -suggestive empirical results have already been obtained.

In this talk we develop a theoretical approach to this Geometric Separation Problem inwhich a deliberately overcomplete representation is chosen made of two frames. One is suitedto pointlike structures (wavelets) and the other suited to curvelike structures (curvelets or shear-lets). The decomposition principle is to minimize the l1 norm of the analysis (rather thansynthesis) frame coefficients. Our theoretical results show that at all sufficiently fine scales,nearly-perfect separation is indeed achieved.

Our analysis has two interesting features. Firstly, we use a viewpoint deriving from mi-crolocal analysis to understand heuristically why separation might be possible and to organizea rigorous analysis. Secondly, we introduce some novel technical tools: cluster coherence,

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rather than the now-traditional singleton coherence and l1 minimization in frame settings, in-cluding those where singleton coherence within one frame may be high.

This is joint work with David Donoho (Stanford University).

2.39 George KyriazisUniversity of Cyprus, Dept. of Mathematics and Statistics, CYPRUS

On the construction of bases and frames for spaces of distributionsWe introduce a new method for constructing bases and frames for general distribution spacesand employ it to the construction of bases for modulation spaces on Rn and frames for Triebel-Lizorkin and Besov spaces on the sphere. Conceptually, our scheme allows the freedom toprescribe the nature, form, or some properties of the constructed basis or frame elements. Forinstance, they can be linear combinations of a small fixed number of shifts and dilates of anysufficiently smooth and rapidly decaying function. On the sphere, our frame elements consistof smooth functions supported on small shrinking caps.

2.40 Richard S. LaugesenDepartment of Mathematics, University of Illinois, UNITED STATES

Frequency-scale frames and the solution of the Mexican hat prob-lem

We resolve a twenty year old open problem on Lp completeness of the time-scale (or wavelet)system generated by the Mexican hat function, when 1 < p < ∞. Our main result concerns“frequency-scale” systems generated by modulation and dilation of a single function. Themixed frame operator (analysis followed by synthesis) is shown to be bijective from Lq to itselfin the frequency domain, for 1 < q < ∞, so that the frequency-scale synthesis operator issurjective onto Lq. Tools include the discrete Calderon condition and a generalization of theDaubechies frame criterion in L2. Completeness of time-scale (wavelet) systems in Lp thenfollows for p ≥ 2, by Fourier imbedding of the frequency-scale systems. For 1 < p < 2, oneproceeds similarly in a Sobolev space in the frequency domain, or in a weighted L2 space inthe time domain. Our criterion applies readily to examples such as the Mexican hat system.

[Joint with H.-Q. Bui.]

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2.41 Tatiana LevitinaTU Braunschweig, GERMANY

Kramer’s sampling theorem and finite Hankel transform eigenfunc-tions

For Hankel–band–limited functions sampling series in terms of the eigenfunctions of the finiteHankel transform are deduced, similar to that obtained by Walter and Shen for Fourier–band–limited functions mainly concentrated on some time interval.

2.42 Franz LuefUC Berkeley, UNITED STATES

Tigth Gabor frames and projections in noncommutative toriOur approach to the construction of projective modules over noncommutative tori in terms ofGabor analysis allows us to link Rieffel’s projections in noncommutative tori with the construc-tion of Gabor frames. The main result demonstrates that Rieffel’s condition on the existenceof projections in noncommutative tori is equivalent to the Wexler-Raz biorthogonality relationsfor tight Gabor frames. Therefore we are able to apply results on the existence of Gabor framesto construct projections in noncommutative tori. The projection associated with a Gabor framegenerated by a Gaussian turns out to be Boca’s projection and the result of Lyubarski and Seipallows us to characterize the range of existence of Boca’s projection.

2.43 Josh MacArthurDalhousie University, CANADA

Wavelets Based on Crystal SymmetriesBaggett et.al. [1], define an internal affine structure on a Hilbert spaceH by a discrete group Γof unitary operators on H and another unitary operator δ on H for which δ−1γδ is an elementof Γ for every γ ∈ Γ. A wavelet relative to the affine structure (δ,Γ) is a finite set {ψ1, . . . , ψn}of vectors inH such that the collection {δj(γ(ψi))} forms an orthonormal basis forH.

Guo et.al. [2], define an affine system with composite dilations by

AAB(Ψ) = {DaDbTkΨ : k ∈ Zn, b ∈ B, a ∈ A},

where Ψ = (ψ1, . . . , ψL) ⊂ L2(Rn), Tk are the translations, Da the dilations, and A,B arecountable subsets of GLn(R). If Ψ, A and B are chosen so that AAB(Ψ) is an orthonormalbasis for L2(Rn), then Guo et.al. call Ψ an AB-multiwavelet.

We reformulate the concept of an affine system with composite dilations [2] in the contextof an internal affine system [1], stressing the underlying representation theory. Then, ratherthan requiring the group to be the semidirect product Zn oB as in [2], we use crystallographic

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groups acting on the plane. By taking R ⊂ R2 to be a ‘particular’ fundamental domain, weshow that it’s characteristic function yields an MRA and corresponding wavelet.

Joint work with Keith Taylor

References[1] Baggett, Larry; Carey, Alan; Moran, William; Ohring, Peter; General existence theorems for orthonormal

wavelets, an abstract approach. Publ. Res. Inst. Math. Sci. 31 (1995), no. 1, 95–111.

[2] Guo, Kanghui; Labate, Demetrio; Lim, Wang-Q; Weiss, Guido; Wilson, Edward; The theory of waveletswith composite dilations. Harmonic analysis and applications, 231–250, Appl. Numer. Harmon. Anal.,Birkhuser Boston, Boston, MA, 2006.

2.44 W. R. MadychUniv. of Connecticut, UNITED STATES

On two issues in analysisAn extension to the multivariate senario of certain theorems concerning the approximation ofband limited functions in terms of their their samples naturally leads to questions regarding thegrowth as |x| goes to infinity of functions u(x) whose derivatives of order k are in Lp(Rn).The null space of such a class of functions consists of polynomials of degree ≤ k − 1 so itis reasonable to expect that the bound on the growth of such functions u(x) should be no lessthan O(|x|k−1) as |x| → ∞. Thus it is somewhat curious that in the case n/p > 1 significantlybetter growth estimates are possible if one is willing to subtract an appropriate polynomial fromu. Roughly speaking one gets growth no greater that O

(|x|k−n/p`(x)

)where `(x) is different

from one only when n/p is an integer, in which case it is a power of log |x| which depends onlyon p.

If {φ(x − k)}k∈Z is an orthonormal basis for a subspace V of L2(R) then the periodicfunction Φ(x) =

∑k∈Z |φ(x−k)|2 plays a role in certain statistical considerations. Of particular

interest are the locations of its maximum value. In the case when V is the subspace of cardinalsplines of order n then Φ(x) is a piecewise polynomial of degree 2(n − 1) whose behaviorappears to be independent of the specific nature of φ. We have completely analyzed the specificcases n = 2, 3, . . . , 7 and can conclude that the corresponding Φ(x) has exactly one maximumon the interval 0 ≤ x

2.45 George Makrakis,

Semiclassical asymptotics of the Wigner equation near causticsWe consider the problem of high-frequency asymptotics for the time-dependent one-dimensional Schr”odinger equation with rapidly oscillating initial data. This problem is com-

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monly studied via the WKB method. An alternative method is based on the limit Wignermeasure. This approach recovers geometrical optics, but, like the WKB method, it fails atcaustics. To remedy this deficiency we employ the semiclassical Wigner function which is aformal asymptotic approximation of the scaled Wigner function but also a regularization of thelimit Wigner measure. We obtain Airy-type asymptotics for the semiclassical Wigner functionas solutions to the Wigner equation.

2.46 Qasem NarimaniUniversity of Mohaghegh Ardabili, IRAN, ISLAMIC REPUBLIC OF

A characterization of closed shift invariant spaces of L2(Rd) and ap-plications

The theory of shift invariant subspaces of L2(Rd) has been studied by several authors dueto its applications in many areas such as wavelet analysis, approximation theory and samplingtheory. The study of shift invariant subspaces in the context of L2(T ) goes back to Beurling andWiener, who have well known results in this topic. We will focus on Wiener’s theorem whichbriefly states that a closed subspaces V of L2(T ) is invariant under the operators of modulationby k(k ∈ Z) iff there exists a measurable subset E of T such that V = L2(E). Identifying thetorus T with [0, 1), and using the Fourier transform on L2(R) we may apply Wieners theoremto characterize closed shift invariant subspaces of L2(R) whose elements are band-limited tosome subset E of [0, 1). In this talk we will use Wieners theorem to characterize closed shiftinvariant subspaces L2(Rd), and we will show that for any closed shift invariant subspace V ofL2(Rd), there exists a sequence Ei (maybe finite) of subsets of T d such that V is isometricallyisomorphic to direct sum of L2(Ei). Some applications and related results will be presented.

2.47 Fabio Nicola,

Boundedness of Fourier intergral operators on Fourer-Lebesguespaces and related topics

2.48 Shahaf Nitzan-HahamovTel Aviv University, ISRAEL

Frame-type systemsFrames, which intuitively (but not quite accurately) are thought of as ”over complete bases inHilbert spaces”, have created much interest over the past 20 years. Unfortunately, there aresome settings in which it is quite difficult, if not imposable, to build a system which on one

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hand satisfies some structural requirements and on the other hand is a frame. For example, it isknown that a family of translates of a single function can never be a frame in L2(R).

In this talk a quasi-frame type of systems will be introduced. Roughly speaking, these arecomplete systems with an additional property - the coefficients of the approximating linearcombinations are, in some sense, controlled. In various situations this concept enables us to getpositive results which are known to be impossible for frames.

Some new directions in this subject, will be discussed at the end of the talk.This is a joint work with A.M.Olevskii.

2.49 Isaac PesensonCCP and TU, UNITED STATES

Eigenmaps and minimal and bandlimited immersions of graphs intoEuclidean spaces

We introduce concepts of minimal immersions and bandlimited (Paley-Wiener) immersions ofcombinatorial weighted graphs (finite or infinite) into Euclidean spaces. The notion of ban-dlimited immersions generalizes the known concept of eigenmaps of graphs. It is shown thatour minimal immersions can be used to perform interpolation, smoothing and approximation ofimmersions of graphs into Euclidean spaces. It is proved that under certain conditions minimalimmersions converge to bandlimited immersions. Explicit expressions of minimal immersionsin terms of eigenmaps are give. The results can find applications for data dimension reduction,image processing, computer graphics, visualization and learning theory.

see also:http://univie.ac.at/nuhag-php/dateien/talks/1103_Strobl2009.pdf

2.50 Pencho PetrushevDepartment of Mathematics University of South Carolina Columbia, SC 29208 USA, UNITED

STATES

Sub-exponentially localized kernels and frames in the context ofclassical orthogonal expansions

Orthogonal polynomials on the the d-dimensional cube, ball, and simplex with weights as wellas spherical harmonics and tensor product Hermite and Laguerre functions are considered. Itshown that the related kernels induced by specially constructed smooth cutoff functions havesub-exponential localization. These kernels are employed for the composition of localizedframes on the respective domains, which in turn are used for characterization of the relatedweighted Triebel-Lizorkin and Besov spaces.

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2.51 Gotz PfanderJacobs University Bremen, Germany, GERMANY

The Bourgain Tzafiri restricted invertibility theorem in infinite di-mensional Hilbert Spaces

The Bourgain-Tzafriri Restricted Invertibility Theorem states conditions under which a Rieszbases can be extracted from an overcomplete system of vectors in finite dimensional spaces. Weextend the result to vector dictionaries in infinite dimensional Hilbert spaces using techniquesdeveloped in the theory of localized frames.

Joint work with Pete Casazza

2.52 Alex PowellDepartment of Mathematics, Vanderbilt University, UNITED STATES

Regularity for complete and minimal Gabor systems on a latticeThe Balian-Low Theorem (BLT) is a strong form of the uncertainty principle for Gabor systemsthat form orthonormal bases or Riesz bases for L2(R). We investigate the Balian-Low theoremin the settings of (1) exact systems, and (2) Schauder bases. We prove a new nonsymmetricallyweighted Balian-Low theorem for Gabor systems that are complete and minimal. We alsodiscuss how Gabor Schauder bases relate to the Balian-Low theorem, and characterize a classof Gabor Schauder bases in terms of the Zak transform and product A2 weights. This is jointwork with Chris Heil.

2.53 Abdumalik RakhimovDepaertment of Applied Mathematics of Tashkent Divition of Moscow University, UZBEK-

ISTAN

On an estimation of eigenfunctions of Schrodinger’s operator in aclosed domain.

Solution of boundary value problems and initial problems in bounded domains leads to theconvergence and sumability problems of eigenfunction expansions in a closed domain. Thisparticular problem requires to estimate eigenfunctions near boundary. Methodology of estima-tion eigenfunctions in compact subsets of the domain is well developed and known (see in [?]). Estimation of eigenfunctions in a closed domains occurs some difficulties near the bound-ary. This difficulties can be avoided if we consider boundary conditions that help to estimateeigenfunctions near the boundary with more accurate values. E.I. Moiseev [?] has proved anestimation of eigenfunctions of the first boundary value problem for the elliptic differential op-erator of second order with smooth coefficients. In the present paper we use methodology ofthe paper [?] and obtain eigenfunction estimations in a closed domain for the elliptic operatorwith singular coefficients.

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2.54 Holger RauhutHausdorff Center for Mathematics, University of Bonn, GERMANY

Circulant and Toeplitz matrices in compressed sensingCompressed sensing seeks to recover a sparse vector from a small number of linear and non-adaptive measurements. While most work so far focuses on Gaussian or Bernoulli randommeasurements we investigate the use of partial random circulant and Toeplitz matrices in con-nection with recovery by `1-minization. In contrast to recent work in this direction we allowthe use of an arbitrary subset of rows of a circulant and Toeplitz matrix. Our recovery resultpredicts that the necessary number of measurements to ensure sparse reconstruction by `1-minimization with random partial circulant or Toeplitz matrices scales linearly in the sparsityup to a log-factor in the ambient dimension. This represents a significant improvement overprevious recovery results for such matrices. As a main tool for the proofs we use a new versionof the non-commutative Khintchine inequality.

2.55 Sivananthan SampathJohann Radon Institute(RICAM), Linz, AUSTRIA

Dual regularized total least squares as a two-parameter regulariza-tion learning algorithm

In sampling problem one seeks to recover a function f : X → R from its samples {f(xi) : xi ∈X}. In practice a sample set is finite and given data may not be exact (noisy), say f(xi) ≈ yi.Then one is interested to recover a function inside the scope of sample set (interpolation) orbeyond the scope of the sample set (extrapolation). Clearly both problems are ill-posed and arecent approach to solve them is based on statistical learning theory and regularization networks[1], where one minimizes the following Tikhonov-type functional

1

n

n∑i=1

(yi − g(xi))2 + λ||g||2H

over a reproducing kernel Hilbert space H generated by a positive definite kernel functionK. Here n is a sample size, and λ is a regularization parameter. The above one parameterregularization method gives satisfactory results for interpolation type problems, but sometimesit fails in extrapolation type problems.

As a remedy, we consider two-parameter regularization method, namely dual regularizedtotal least square method [2], [3]. In the present context this method reconstructs a function asthe minimizer of the problem

||S∗xSxg − S∗xy||2H + α||Bg||2H + λ||g||2H → min,

where Sx : H → Rn is sampling operator defined by Sxg = (g(xi))ni=1, S∗x : Rn → H is

adjoint of Sx, B =√n|Sx| =

√n(S∗xSx)

12 , y = (y1, y2, . . . , yn), and ||y||2Rn = 1

n

∑ni=1 y

2i . We

show that the method gives satisfactory results for extrapolation type problems, and apply it fora prediction of the blood glucose concentration from past measurements.

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References[1] T. Evgeniou, M. Pontil and T. Poggio, Regularization networks and support vector ma-

chines, Adv. Comp. Math., 13:1-50, 2000.

[2] Shuai Lu, S. Pereverzev and U. Tautenhahn, A model function method in total leastsquares, RICAM-Report No. 2008-18, 2008.

[3] Shuai Lu, S. Pereverzev and U. Tautenhahn, Dual regularized total least squares and multi-parameter regularization, Computational Methods in Applied Mathematics, 8(3):253-262,2008.

2.56 Myung-Sin SongSouthern Illinois University Edwardsville, UNITED STATES

Spectral Theory of Discrete ProcessesIn this talk we discuss spectral analysis for a class of transfer operators. These transfer operatorsarise for a wide range of stochastic processes, ranging from random walks on infinite graphsto the processes that govern signals and recursive wavelet algorithms; even spectral theory forfractal measures. In each case, there is an associated class of harmonic functions which westudy. The following questions will be addressed:

In specific applications, and for a specific stochastic process, how do we realize the transferoperator T as an operator in a suitable Hilbert space? And how to spectral analyze T once theright Hilbert space H has been selected? Finally the stochastic processes that are governed bya single transfer operator can be characterized.

This circle of problems is both interesting and non-trivial as it turns out that T may often bean unbounded linear operator inH; but even if it is bounded, it is a non-normal operator, so itsspectral theory is not amenable to an analysis with the use of von Neumann’s spectral theorem.

2.57 Diana StoevaUniversity of Architecture, Civil Engineering and Geodesy, BULGARIA

Perturbation of frames in Banach spacesIn this talk we consider perturbation of Xd-Bessel sequences, Xd-frames, Banach frames,atomic decompositions and Xd-Riesz bases in separable Banach spaces. Equivalence betweensome perturbation conditions is analyzed.

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2.58 Joachim ToftVaxjo University, SWEDEN

Wave-front set of Fourier Lebesgue and modulation space typesRoughly speaking, a wave-front set WF∗(f) of the distribution f with respect to ”something”,give information where the distribution f has singularities with respect to this ”something”,as well as what directions in these points of singularities, the singularities propagates. Forexample, assume that this ”something” is smoothness (i. e. it has singularity at x0 ∈ Rd ifit is not infinitely many times differentiable around this point), and consider the distributionf ∈ D ′(R2), given by

f(x, y) = H(y),

where H(y) is the Heaviside function, which is equal to 1 when y ≥ 0 and equal to 0 when y

2.59 Tino UllrichHausdorff-Center for Mathematics, Bonn, GERMANY

Coorbit space theory for inhomogeneous spaces of Besov-Triebel-Lizorkin type

We study coorbit space theory for inhomogeneous spaces of Besov-Triebel-Lizorkin type withdominating mixed smoothness. These spaces turn out to be coorbit spaces in a certain sense.Applying an abstract machinery we obtain useful atomic decompositions by discrete framesand wavelets.

2.60 Vladimir VasilyevBryansk State University, RUSSIAN FEDERATION

Discrete singular integrals and related symbolic calculusWe consider for simplicity two dimensional case. Given convolution singular Calderon-Zygmund integral

(Au)(x) = v.p.

∫R2

K(x− y)u(y)dy, x ∈ R2, (1)

which is bounded operator L2(R2) → L2(R2), and let σ(ξ) be its symbol(smooth) defined onunit circle S1.

We can write instead of operator (1) its discrete analogue

(Au)(x) =∑y∈Z2

K(y)u(x− y), x ∈ Z2, (2)

where Z2 is integer point lattice in R2, K is a restriction of kernel K on Z2, and by definitionK(0) = 0.

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Denote R2+ the half-plane {x ∈ R2 : x = (x1, x2), x2 > 0}, R2

++ the first quadrant{x ∈ R2 : x = (x1, x2), x1 > 0, x2 > 0}, and respectively P+, P++ projectors on R2

+,R2++.

The equation with operator (1) is easily solvable by Fourier transform. For the equations

P+AP+u+ = f+ (1′)

P++AP++u++ = f++ (1′′)

in general solvability situations are described from different points of view. For studying (1′′)the author used earlier the special (wave) factorization of symbol σ(ξ) related to Fourier imageof projector P++[1]. Here we will try to describe the Fourier image of discrete analogues ofprojectors P+, P++ for studying analogues of equation with operator (2) (namely (2′), (2′′)).

1. V.B. Vasilyev, Fourier multipliers, pseudo differential equations, wave factorization,boundary value problems, Moscow, Editorial URSS, 2006 (in Russian).

see also:http://univie.ac.at/nuhag-php/dateien/talks/1251_strobl09.pdf

2.61 Apostolos VourdasUniversity of Bradford, UNITED KINGDOM

Totally disconnected and locally compact Heisenberg-Weyl groupsHarmonic analysis on Z(p`) and the corresponding representation of the Heisenberg-Weylgroup HW [Z(p`),Z(p`),Z(p`)], is studied. It is shown that the HW [Z(p`),Z(p`),Z(p`)]with a homomorphism between them, form an inverse system which has as inverse limitthe profinite representation of the Heisenberg-Weyl group HW[Zp,Zp,Zp]. Harmonic anal-ysis on Zp is also studied. The corresponding representation of the Heisenberg-Weyl groupHW[(Qp/Zp),Zp, (Qp/Zp)] is totally disconnected and locally compact topological group.

see also:http://univie.ac.at/nuhag-php/dateien/talks/1064_slidestro.pdf

2.62 Jan VybiralFriedrich-Schiller University, Jena, GERMANY

Traces of radial Sobolev, Besov and Triebel-Lizorkin spacesWe study the radial subspaces of Sobolev, Besov and Triebel-Lizorkin spaces. We describedecay properties of these functions and their traces. We use classical techniques in the case ofSobolev spaces and atomic decomposition to deal with Besov and Triebel-Lizorkin spaces.

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2.63 Ferenc WeiszDepartment of Numerical Analysis, Eotvos University, HUNGARY

Gabor expansions and local Hardy spacesThe reconstruction and coefficient operators of Gabor expansions and Walnut representationare investigated in local Hardy spaces. It is proved that the Hardy norm of f is equivalent tocertain norm of the Gabor coefficients. New Wiener amalgam spaces are introduced for localHardy spaces. A general summability method, the so called θ-summability is considered forGabor series. It is proved that the maximal operator of the θ-means is bounded from hp to Lpand from the amalgam space W (hp, `∞) to W (Lp, `∞). This implies the almost everywhereconvergence of the θ-means for all f ∈ W (L1, `∞).

2.64 Piotr WojdyłłoInstitute of Mathematics, Polish Academy of Sciences, POLAND

Quantum Approach for Acoustics ReconstructionThe algorithms used for the reconstruction of acoustical enclosures involve either via solutionof diffusion equation or some variants of ray tracing techniques. In the latter case, to incor-porate the effects of the material the coefficients concerning absorption effect are introducedin approaches based on Lambert’s law, while in the former appropriate boundary conditionsdependent on the expected acoustical properties of the material are used. The still more com-plex reflection/absorption model we present for the use in ray tracing algorithm would involvefurther material properties even to the level of material microstructure related so to quantum ef-fects. Although, they are usually considered immaterial for the problem at this wavelength anddistances, the effects of timbre modification might be related to highly localized phenomena intime-frequency plane and so involve the quantum effects of the material structure.

The application of the method would impose opportunities to manipulate the timbre of therecording by the suitable adjustment of interior via which it is processed. It would further allowa research on the subjective quality of the recording by the use of interiors known of their greatacoustical properties.

As for such interiors long enough reverberations are a must, we outsource to 5 secondstime of such which involves heavy computational load at CD quality. The quantum approachalgorithm to be presented is relatively slow in terms of time needed for simulation from thegeometrical and material information of the enclosure but may be applied in the situation ofvirtual acoustical visits, for instance.

The speed-up effects of the processing itself allow the offline processing in time shortherthan that of recording and short recordings (up to 6 minutes) to be played from memory on-lineimplementing the equivalent of 5 seconds reverberation at 44.1 KHz sampling in 30% of theload for Quad Core processor with 2GB RAM.

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REFERENCES 35

2.65 Georg ZimmermannInstitut fur Angewandte Mathematik und Statistik, Universitat Hohenheim, Stuttgart, GER-

MANY

A new proof of the Gasca-Maeztu Conjecture for n = 4

In the Chung-Yao construction of poised nodes for bivariate polynomial interpolation, the inter-polation nodes are intersection points of some lines. The Berzolari-Radon construction is moregeneral, since in this case the nodes of interpolation lie (almost) arbitrarily on some lines. In1982 Gasca and Maeztu conjectured that every poised set allowing the Chung-Yao constructionis of Berzolari-Radon type. So far, this conjecture is confirmed only for polynomial spaces ofsmall total degree n ≤ 4, the result being evident for n ≤ 3. In case n = 4 two proofs areknown: one of J. R. Busch, and another of J. M. Carnicer and M. Gasca. Here we present athird proof which seems to be more geometric in nature and perhaps easier. If time allows, wealso present some results for the case of n = 5 and for general n which might be useful forlater consideration of the problem.

3 Posters

3.1 Ivana CarrizoNuHAG, ARGENTINA

Minimizers of a mixed frame potentialWe define a mixed frame potential of a finite dimensional Hilbert space and characterize itsminima on a restricted domain. We obtain necessary and sufficient conditions on a real se-quence {cm}m=1,...,N in order to have a dual pair of frames {fm}m=1,...,N , {gm}m=1,...,N suchthat 〈fm, gm〉 = cm.This is joint work with Sigrid Heineken.

3.2 Ioan GolePolitehnica Unersity of Timisoara, ROMANIA

ON SAMPLING AND APPROXIMATION OF RANDOM SIGNALSON SAMPLING AND APPROXIMATION OF RANDOM SIGNALS

Ioan Golet, Department of MathematicsRandom signals are used in solving a large class of problems (there are sound signals ,

optical signals, electronic signals, impulses, noises, and so on). In the last time the study ofrandom signals has became object of multidisciplinary research. An important progress intelecommunications, in diagnosis systems is due to the study of random signals. Generallyspeaking, by a random signal defined on a time set T with values in a separable Banach spaceX, we mean a family of random variables with values into a Banach space X, indexed under atime parameter from a time set T. The set of all random variables on the same base probabilitymeasure space can be endowed with a structure of probabilistic normed space. Then, a oneto one correspondence: a random signals to a function with values into probabilistic normedspace can be stated. In virtue of this correspondence we will use probabilistic analysis methodsto study properties of random We show that framework of probabilistic normed spaces givessome new possibilities to study some properties of random signals. The main results of thepaper are the approximations theorems of the continuous random signals by using probabilisticdistribution functions at selected time moments and sampling methods

3.3 Sigrid Bettina HeinekenNuHAG, AUSTRIA

Minimizers of a mixed frame potentialWe define a mixed frame potential of a finite dimensional Hilbert space and characterize itsminima on a restricted domain. We obtain necessary and sufficient conditions on a real se-

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quence {cm}m=1,...,N in order to have a dual pair of frames {fm}m=1,...,N , {gm}m=1,...,N suchthat 〈fm, gm〉 = cm. This is joint work with Ivana Carrizo.

3.4 Srdan MaksimovicFaculty of Maritime Studies, University of Rijeka, CROATIA

From scaling sets to scaling functionsWe present a general method of constructing scaling functions in Rn for an arbitrary expandingmatrix with integer coefficients. Using a scaling set as a starting point, values of the corre-sponding characteristic function are modified in a way that obtained object still remains theFourier transform of a scaling function. Moreover, it is shown that every MRA wavelet can beconstructed using this procedure.

3.5 Serap OztopIstanbul University, TURKEY

A Note on Multipliers of weighted Lp(G,A) spacesLet G be a locally compact group, 1 < p < 1: The aim of this paper is to characterize themultipliers of the weighted Banach valued intersection Lp(G) spaces as the space of multipliersof certain Banach algebra.

see also:http://univie.ac.at/nuhag-php/dateien/talks/1213_somemultipliers.pdf

3.6 Georg RieckhAW, AUSTRIA

Wavelets and Frames for Acoustic BEM.We are presenting the Boundary element method (BEM) as a means of solving problems oc-curring in the field of acoustics. The poster will provide an overview of the relevant literatureand give an outlook on my dissertation project including ideas from the fields of wavelets andframes.

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3 POSTERS 38

3.7 Jose Luis RomeroBuenos Aires University, ARGENTINA

Explicit localization estimates for dual framesGiven a polynomially self-localized frame, we derive some explicit estimates for the localiza-tion of the dual frame. The estimates do not show the full preservation of the original local-ization conditions but are completely explicit. This yields some qualitative consequences forspline-type spaces.