Time-Frequency Analysis Meets Quantum Harmonic Analysis
Abstract
The subject of this thesis is the study of quantum harmonic analysis and timefrequency analysis, and in particular the intersection of these two fields. Quantum harmonic analysis is studied abstractly both by obtaining new results and by extending the setting to other abelian and nonabelian groups. Tools and results from quantum harmonic analysis are used to study concepts from time-frequency analysis, for instance localization operators and Cohen’s class, obtaining new results and generalizations and reinterpretations of old results in time-frequency analysis. Concepts and results in time-frequency analysis also inspire new directions, results and proofs in quantum harmonic analysis such as the careful study of Fourier series of operators in a general setting. Sammendrag
I denne avhandlingen studeres de to matematiske teoriene kvante-harmonisk analyse og tid-frekvens-analyse, med et spesielt fokus pa skjaringspunktet mellom disse teoriene. Vi studerer kvante-harmonisk analyse abstrakt, bade ved a vise nye resultater og gjennom a utvide domenet hvor kvante-harmonisk analyse er gyldig til andre abelske og ikke-abelske grupper. I tillegg bruker vi redskaper og resultater fra kvante-harmonisk analyse til a studere konsepter i tid-frekvens-analyse, som lokaliseringsoperatorer og Cohens klasse, og finner derigjennom bade nye resultater samt generaliseringer og nytolkninger av gamle resultater i tid-frekvens-analyse. Konsepter og resultater i tid-frekvens-analyse inspirerer ogsa nye retninger, resultater og bevis i kvante-harmonisk analyse, eksempelvis en grundig studie av Fourierrekker for operatorer.
Has parts
Paper A: Luef, Franz; Skrettingland, Eirik. Mixed-State Localization Operators: Cohen’s Class and Trace Class Operators. Journal of Fourier Analysis and Applications 2019 ;Volum 25.(4) s. 2064-2108 https://doi.org/10.1007/s00041-019-09663-3Paper B: Luef, Franz; Skrettingland, Eirik. On Accumulated Cohen's Class Distributions and Mixed-State Localization Operators. Constructive approximation 2019 ;Volum 52.(1) s. 31-64 https://doi.org/10.1007/s00365-019-09465-2
Paper C: Skrettingland, Eirik. Quantum Harmonic Analysis on Lattices and Gabor Multipliers. Journal of Fourier Analysis and Applications 2020 ;Volum 26.(3) https://doi.org/10.1007/s00041-020-09759-1 This article is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0)
Paper D. Skrettingland, Eirik. On Gabor g-frames and Fourier Series of Operators. Studia Mathematica, 2021, volume 259, issue 1. https://doi.org/ 10.4064/sm191115-24-9
Paper E: Luef, Franz; Skrettingland, Eirik. A Wiener Tauberian theorem for operators and functions. Journal of Functional Analysis 2021 ;Volum 280.(6) https://doi.org/10.1016/j.jfa.2020.108883 This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0)
Paper F: Skrettingland, Eirik. Equivalent Norms for Modulation Spaces from Positive Cohen’s Class Distributions
Paper G: Berge, Eirik; Berge, Stine Marie; Luef, Franz; Skrettingland, Eirik. Affine Quantum Harmonic Analysis