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.

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