Projektdetails
Abstract
Wider research context / theoretical framework:
This proposal concerns coherent states and frames arising from unitary representations of Lie groups. The systematic investigation of coherent frames has mainly been divided into the concrete cases of the Heisenberg and affine group in time-frequency and wavelet analysis, respectively, where they provide powerful tools for studying pseudodifferential and singular integral operators in classical Fourier analysis.
However, when developing analogous methods for multiparameter Fourier analysis, it is essential to use frames that are compatible with multiparameter dilation structures. This motivates the study of coherent frames of more general exponential Lie groups. Despite various recent developments, many key problems regarding coherent frames are still open and several phenomena are not well understood. Additionally, while multiparameter singular integrals and anisotropic wavelets have individually received considerable attention in recent years, the relationship between these topics is still largely unexplored.
Hypotheses / research questions / objectives:
This project aims to develop a systematic theory of coherent frames that is significant and specific for exponential Lie groups. In connection with this, various questions in the representation theory of exponential Lie groups will be addressed. Finally, the developed techniques will be used for a wavelet theoretic analysis of multiparameter singular integrals. Specifically, the key objectives are to study:
O1. Asymptotic properties of matrix coefficients on exponential Lie groups.
O2. Density conditions for coherent frames of exponential Lie groups.
O3. Almost diagonalisation of multiparameter singular integrals.
The project contributes to these objectives by studying a set of six research problems.
Approach / methods:
The approach towards objectives O1, O2 and O3 is based on a combination of techniques from harmonic analysis, ranging from Fourier analysis to the representation theory of solvable Lie groups. Particularly important notions and techniques include: coadjoint orbits and Fourier inversion on nilpotent Lie groups (O1), Beurling-type densities and quasi-lattices (O2) and smooth molecular decompositions of multiparameter singular integrals (O3).
Level of originality / innovation:
The proposed project will be the first systematic study on density conditions for coherent frames arising from possibly non-square-integrable representations of non-unimodular groups. Additionally, it is the first systematic investigation of wavelet methods in the study of multiparameter singular integrals, with the aim of bridging the gap between the active fields of anisotropic wavelets and multiparameter singular integrals. The expected results will strengthen the connection between Euclidean Fourier analysis and representation theory.
Primary researchers involved:
J.T. van Velthoven (PI), M. Bownik, H. Führ, V. Oussa
This proposal concerns coherent states and frames arising from unitary representations of Lie groups. The systematic investigation of coherent frames has mainly been divided into the concrete cases of the Heisenberg and affine group in time-frequency and wavelet analysis, respectively, where they provide powerful tools for studying pseudodifferential and singular integral operators in classical Fourier analysis.
However, when developing analogous methods for multiparameter Fourier analysis, it is essential to use frames that are compatible with multiparameter dilation structures. This motivates the study of coherent frames of more general exponential Lie groups. Despite various recent developments, many key problems regarding coherent frames are still open and several phenomena are not well understood. Additionally, while multiparameter singular integrals and anisotropic wavelets have individually received considerable attention in recent years, the relationship between these topics is still largely unexplored.
Hypotheses / research questions / objectives:
This project aims to develop a systematic theory of coherent frames that is significant and specific for exponential Lie groups. In connection with this, various questions in the representation theory of exponential Lie groups will be addressed. Finally, the developed techniques will be used for a wavelet theoretic analysis of multiparameter singular integrals. Specifically, the key objectives are to study:
O1. Asymptotic properties of matrix coefficients on exponential Lie groups.
O2. Density conditions for coherent frames of exponential Lie groups.
O3. Almost diagonalisation of multiparameter singular integrals.
The project contributes to these objectives by studying a set of six research problems.
Approach / methods:
The approach towards objectives O1, O2 and O3 is based on a combination of techniques from harmonic analysis, ranging from Fourier analysis to the representation theory of solvable Lie groups. Particularly important notions and techniques include: coadjoint orbits and Fourier inversion on nilpotent Lie groups (O1), Beurling-type densities and quasi-lattices (O2) and smooth molecular decompositions of multiparameter singular integrals (O3).
Level of originality / innovation:
The proposed project will be the first systematic study on density conditions for coherent frames arising from possibly non-square-integrable representations of non-unimodular groups. Additionally, it is the first systematic investigation of wavelet methods in the study of multiparameter singular integrals, with the aim of bridging the gap between the active fields of anisotropic wavelets and multiparameter singular integrals. The expected results will strengthen the connection between Euclidean Fourier analysis and representation theory.
Primary researchers involved:
J.T. van Velthoven (PI), M. Bownik, H. Führ, V. Oussa
| Kurztitel | Kohärenter frames |
|---|---|
| Status | Laufend |
| Tatsächlicher Beginn/ -es Ende | 1/09/24 → 31/08/28 |