Jaak Peetre, Centre for Mathematical Sciences, Lund University, Sweden
Michael Cwikel, Department of Mathematics, Technion - Israel Institute of Technology, Haifa, Israel
Of course, in his talk, Jaak will be telling us plenty of things about his wonderful students Annika and Gunnar. But I think that I also have a rather special connection with the Sparr family. On three occasions I thought that I had obtained a very nice new result, and I even published two of those results. However it turned out that either Annika or Gunnar had already obtained these same results, and in fact more general versions of them. I will talk about various subsequent extensions of a number of results of Gunnar and of Annika. Some of these are not very new, but one of them, concerning convexifications of Calderón couples, will be announced for the first time in this talk. My work with Charles Fefferman on the canonical seminorm for Weak L1 was a followup to Annika's remarkable discovery that Weak L1 has a non trivial dual. Charlie was quite surprised and interested when he first heard of that discovery. Apparently I will not have time to talk about all the projects in which I have been involved which are offshoots in one way or another of some work of Gunnar or of Annika. But, to give you some very partial indication about the magnitude of the ``Sparr impact factor'', let me simply list my coauthors in those projects: Yacin Ameur, Jonathan Arazy, Ronald Coifman, Charles Fefferman, Svante Janson, Björn Jawerth, Uri Keich, Inna Kozlov, Mieczyslaw Mastylo, Mario Milman, Per
Lech Maligranda
Results from two most cited papers of Gunnar Sparr in the interpolation theory (published in 1974 and 1978) and results of Annika Haaker (Sparr) on Lorentz spaces will be presented. Then some recent results in these connections will be described and explained, including those from my joint papers.
References
[G1] G. Sparr, Interpolation of several Banach spaces, Ann. Mat. Pura Appl. (4) 99 (1974), 247--316.
[G2] G. Sparr, Interpolation of weighted L_{p } -spaces, Studia Math. 62 (1978), 229--271.
[A1] A. Haaker, On the conjugate space of Lorentz space, Technical Report, Lund 1970, 1--23.
[A2] A. Sparr, On the conjugate space of the Lorentz space L(φ,q), Contemporary Mathematics, 445 (007), 313--336.
[AKMNP] I. Asekritova, N. Krugljak, L. Maligranda, L. Nikolova and L. E. Persson,Lions-Peetre reiteration formulas for triples and their applications, Studia Math. 145 (2001), no. 3, 219--254.
[CKMP] M. Cwikel, A. Kamińska, L. Maligranda and L. Pick, Are generalized Lorentz "spaces " really spaces?, Proc. Amer. Math. Soc. 132 (2004), no. 12, 3615--3625.
[KM1] A. Kamińska and L. Maligranda, Order convexity and concavity of Lorentz spaces Λ_{p,w}, 0<p<∞, Studia Math. 160 (2004), no. 3, 267--286.
[KM2] A. Kamińska and L. Maligranda, On Lorentz spaces Γ_{p,w}, Israel J. Math. 140 (2004), 285--318.
[KrM] A. Karlovich and L. Maligranda, On the interpolation constant for Orlicz spaces, Proc. Amer. Math. Soc. 129 (2001), no. 9, 2727--2739.
[M] L. Maligranda, Type, cotype and convexity properties of quasi-Banach spaces, In: Banach and Function Spaces, Yokohama Publ., Yokohama 2004, 83--120.
[MO] L. Maligranda
and V. I. Ovchinnikov, On interpolation between L^{1}+L^{∞}
and L^{1}∩L^{∞},
J. Functional Anal. 107 (1992), 342--351.
Lars Erik Persson, Department of Mathematics, Luleå University of Technology, Sweden
I will begin by presenting some prehistory of the dramatic time before Hardy finally presented and proved his inequality in 1925 (see [2]). After that I will present some pers(s)onal reflections concerning the almost unbeleivable further developments and applications of this type of inequalities (see e.g. the books [1] and [3] and the references given there). Moreover, I will shortly mention some other even newer and very surprising results together with a number of open questions these results have generated. Finally , I will mention an other Hardy related inequality by Gunnar Sparr and also some other surprises in this connection.
References
[1] Alois Kufner and Lars-Erik Persson, World Scientific, New Jersey/London/ Singapore/Hong Kong, 2003.
[2] Alois Kufner, Lech Maligranda and Lars-Erik Persson, The prehistory of the Hardy inequality, Amer. math. Monthly 113 (2006), No. 8, 715-732.
[3] Alois Kufner, Lech Maligranda and Lars-Erik Persson, The Hardy Inequality. About its history and some related results, Vydavatelsky Servis Publishing House, Pilsen, 2007.
Frank Hansen, Department of Economics, Copenhagen University, Danmark
Sten Kaijser, Department of Mathematics, Uppsala University, Sweden
About 10 years ago one of my students, Yacin Ameur, was working on interpolation of Hilbert spaces. The starting point was a paper by W. Donoghue from 1968. This was a beautiful paper but it was written at a time before the K-functional had come to completely dominate interpolation theory. This implied in particular that the results had to be translated into "modern terminology". In a beautiful thesis Yacin was able to give a new proof of Donoghue's result, as well as proving that Hilbert couples are Calderon couples. For the participants of this conference it will not come as a surprise that the key ideas in Yacin's work came from two of Gunnar Sparr's papers. I would like to point out that interpolation of Hilbert spaces is a very amusing topic, mainly because there are so many inner products around. This means that it is necessary to choose one particular inner product as "the inner product", and this inner product is then used to define "adjoint operators". The other inner products will then be expressed in terms of positive operators, and more important, adjoints with respect to these will also be expressed in terms of these operators. One of the possible choices for "the inner product" inspired me to consider some Hilbert spaces of functions on the entire real line, as well as analytic functions in a strip. As a consequence I "discovered" some interesting sets of orthogonal polynomials and this in turn led to a thesis by another student Tsehaye K. Araaya. In my talk I will present the work by Yacin Ameur and Tsehaye K. Araaya.
Anders Rantzer,
We will discuss some current research devoted to approximation of dynamical system for use in engineering. Of particular interest is the problem to approximate an analytic function on the unit disc with a rational function of low order. Computational methods have been developed using linear matrix inequalities and convex optimization.We will describe some recent methods and relate them to challenges posed by Toyota Motor Company. Strikingly enough, some of the main tools are related to mathematical topics that Gunnar brought to my attention when he acted as supervisor for my first years as PhD student 20 years ago.
Bo Bernhardsson,
Some mathematical modeling problems - 20 years later In the middle of the 80s I had the pleasure of being a student at the mathematics department with Gunnar as one of my teachers. This was before he had started to focus on the image analysis area. I will talk about several interesting problems in applied mathematics he introduced me to at that time and what has happened to these. This will show his fine ability to find fruitful opportunities for mathematical modeling and analysis in engineering. I will also show why matrix monotone functions are important for the optimization of future mobile communication systems.
Jan J. Koenderink Delft University of Technology & Universiteit Utrecht
Abstract: In his habilitation lecture Riemann mentions "the space we move in" and "color space" as the only multiply extended continua we know from experience. He probably based this on work by Maxwell, Grassmann and Helmholtz. Colorimetry has important applications because it describes the input to the brain and thus is the link between physics and psychology, the foundation of any "Color Science". Unfortunately the formal structure of colorimetry as current today is very muddled. I will discuss how to "do colorimetry right".
Roger Mohr, Grenoble University
Abstract: During these last ten years huge progress has been accomplished in automating the labelling of images attracting a large interest from the research community as from companies. This talk will present some of the recent results obtained and discuss how the major achievements have been obtained. Mathematical ideas were interleaved with good engineering; complexity had to be handled through smart programming, statistics were integrated in order to absorb the large amount of outliers in this kind of data. From there, it will also point out that a huge amount of work remains to be done in order to approach the performance of the human visual system.
Jan Olof Eklundh, CVAP, KTH, Stockholm, Sweden,
Computer vision as a field has undergone a number of shifts in focus and goals during its more than fifty years of existence. In his foreword to the book Computer Vision (originally published as AI Journal, Volume 17) Mike Brady 1981 wrote: "..Computer Vision(s).....current concentration (is) on topics corresponding to identifiable modules in the human visual systems". In the talk I will discuss how computer vision research has developed from its early days and how it relates to this statement today. I will give illustrations from the work in my own group and also give a number of examples of my interactions with Gunnar since around 1985, some of a technical nature, others on a more personal note.
Anders Heyden,
The talk will cover Gunnars first contributions within image analysis and computer vision, especially linear shape spaces and its connection to Chasles theorem, the Kruppa equations and the Demazure theorem. The thoery of linear shape spaces will be outlined and used to prove the classical Chasles theorem, i.e. the minimal reconstruction problem of seven points in two uncalibrated cameras. We will also give a proof of the minimal reconstruction problem of six points in three uncalibrated cameras based on the theory of shape. Finally, similar methods will be used to prove the Damazure theorem, i.e. the minimal reconstruction problem of five points in two calibrated cameras.
Rikard Berthilsson,
A central problem in computer vision is to reconstruct a scene given a set of images taken from it. By combining the images it is possible to compute the three dimensional structure of the scene together with the positions of the camera. Sparr pioneered the field by introducing the concept of shape. By using shape the reconstruction problem could be formulated as an optimization problem of aligning subspaces in a vector space. The notation of shape, initially used for finite sets, could easily be extended to curves and was later used commercially for handwriting recognition.
Senast uppdaterad: 2008-04-29