Centre for Advanced Study

at the Norwegian Academy of Science and Letters

Operator Related Function Theory and Time-Frequency Analysis

Abstract

The project is devoted to two central fields of modern mathematical analysis, operator related function theory and time–frequency analysis, and the profound interplay between these two areas, which to a large extent are mutually complementary. Complex analysis, which in the first half of the previous century was mainly a purely theoretical discipline, is now a powerful tool in harmonic and functional analysis, probability theory as well as in applied areas such as, for example, control theory and information theory. Time–frequency analysis originated within quantum mechanics and signal analysis; it has grown into an independent mathematical discipline, intertwined with parts of harmonic analysis, combinatorial and geometrical analysis, representation theory, pseudo-differential operators, and C*-algebras. Methods, approaches, and – perhaps even more importantly – the philosophy of time – frequency analysis allow one to reexamine known results, discover new unexplored areas in classical function theory, and also to establish surprising and profound relations between problem arising in areas of mathematics that are seemingly distant from each other. One of the most fascinating examples of such interaction is the so-called Feichtinger conjecture, which arose from problems in time–frequency analysis. A few years ago, the Feichtinger conjecture was shown to be equivalent to longstanding unresolved fundamental conjectures in combinatorial analysis (the paving conjecture) and mathematical physics (the Kadison–Singer conjecture). One possible way of attacking this problem is based on methods from operator related function theory, such as spaces of analytic functions and sampling theory. Another example of similar remarkable interaction between subfields of mathematical analysis is found in scattering theory, which combines methods of mathematical physics, operator related function theory, partial differential equations, and harmonic analysis. This field originated in the study of basic physical phenomena and finds direct applications for example in nanotechnology. The project is aimed to promote collaboration and interaction on these and other similar topics between experts in operator related function theory and time–frequency analysis, with a view to the relevance for engineering and physical sciences. Such interplay will enrich both disciplines and lead to a deeper understanding of important challenges, including the two topics mentioned above. More specifically, the project is expected to lead to new advances in areas such as spaces of analytic functions, spectral function theory, potential theory, Gabor analysis, as well as in applications to scattering theory (in particular, scattering on quantum graphs), partial differential operators, and geometrical analysis.

End Report

The CAS program offered the participants an exceptional opportunity to concentrate full time on hard problems and to open up new directions of research. Collaboration with a number of foreign institutions has been strengthened, and new partnerships have been developed for ongoing and future joint work.

The progress achieved by the program can roughly be divided into three categories:

  • Completion and advancement of projects already initiated and commenced before the program at CAS. A distinguished example of this kind of work is Lacey’s solution to the problem of two weight inequalities for the Hilbert transform. It has been open for three decades and was finally settled during the CAS program. Its solution opens up new perspectives in operator related function theory and is directly linked to fundamental problems about Toeplitz operators and model subspaces. Another example is Bondarenko’s remarkable solution to a question posed by David Larman in the 1970s about Borsuk’s conjecture for two-distance sets. The program provided a stimulating atmosphere to reach these remarkable results and many others documented in the list of publications.
  • Solution to problems that surfaced during the program. Examples of such kind of projects are the work of Queffélec and Seip on approximation numbers of composition operators, the sharp estimates of Aistleitner, Berkes, and Seip for certain GCD sums, leading to a Carleson-Hunt inequality for systems of dilated functions of bounded variation, the work of Aleman, Lyubarskii, Malinnikova, and Perfekt, which contains a full description of Schatten classes for composition operators on model spaces, and new results of Gröchenig and Lyubarskii on multidimensional Gabor frames. The list of publications reflects a considerable interaction between the many researchers that took part in the CAS program.
  • Initial steps in the development of future projects. This output of the CAS program is not documented in the list of publications, but is nevertheless an exceptional and invaluable long-term effect of the CAS program. While we cannot speak for all the participants in the program, we have recorded the following list of emerging collaborative projects for the core team of NTNU researchers: Seip is involved in projects with Berkes, Bondarenko, Hilberdink, Montes, Queffélec, Sehba, and Weber; Lyubarskii and Malinnikova are involved in projects with Aleman, Luef, Perfekt, Putinar, Dörfler, Mozolyako, and Nicolau. These new projects cross many traditional borders between mathematical subfields and intertwine subjects such as probability theory, analytic number theory, operator theory, complex analysis, potential theory, harmonic analysis, and time-frequency analysis. The broad interaction and stimulating atmosphere within the CAS program have been crucial for opening up these new research directions.

The impetus of the CAS program can be expected to be extensive for many years to come; it will be essential for the success of future projects, such as the remaining tasks of two ongoing projects supported by the Research Council of Norway: “Discrete Models in Mathematical Analysis” (2012 – 2015), a FRIPRO project headed by Yurii Lyubarskii and “Dirichlet Series and Analysis on Ploydiscs” (2013 – 2018), a FRIPRO project realizing an ERC Advanced Grant application, headed by Kristian Seip.

Fellows

  • Belov, Yurii
    Postdoctoral Fellow St.Petersburg State University 2012/2013
  • Borichev, Alexander
    Professor Aix-Marseille University 2012/2013
  • Eikrem, Kjersti Solberg
    Ph. D. Candidate Norwegian University of Science and Technology (NTNU) 2012/2013
  • Grepstad, Sigrid
    Dr. Norwegian University of Science and Technology (NTNU) 2012/2013
  • Gröchenig, Karlheinz
    Dr. University of Vienna 2012/2013
  • Lacey, Michael Thoreau
    Professor Georgia Institute of Technology 2012/2013
  • Malinnikova, Eugenia
    Associate Professor Norwegian University of Science and Technology (NTNU) 2012/2013
  • Montes Rodriguez, Alfonso
    Professor University of Seville 2012/2013
  • Ortega Cerdá, Joaquim
    - University of Barcelona 2012/2013
  • Perfekt, Karl-Mikael
    Ph. D. Candidate Lund University 2012/2013
  • Poltoratski, Alexei
    Professor Texas A&M University 2012/2013
  • Queffelec, Martine
    Assistant Professor Lille University of Science and Technology 2012/2013
  • Queffelec, Hervé
    Professor Em. Lille University of Science and Technology 2012/2013
  • Saksman, Eero
    Professor University of Helsinki 2012/2013
  • Sodin, Mikhail
    Professor Tel Aviv University 2012/2013
  • Vähäkangas, Antti
    Postdoctoral Fellow University of Helsinki 2012/2013
  • Weber, Michel Jeon Georges
    Director University of Strasbourg 2012/2013

Previous events

Group leader

  • Yurii Lyubarskii

    Title Professor Institution Norwegian University of Science and Technology (NTNU) Year at CAS 2012/2013
  • Kristian Seip

    Title Professor Institution Norwegian University of Science and Technology (NTNU) Year at CAS 2012/2013
LOGO