Course content

Key concepts and results from modern harmonic analysis and time-frequency analysis, including various extensions of Fourier analysis. The content may vary from year to year, and current topics include:

• Modern harmonic analysis: Maximal functions, Calderón-Zygmund decompositions, the Hilbert transform. Littlewood-Paley theory, Hardy spaces, Carleson measures, Cauchy integrals, and singular integral operators. Probabilistic techniques in harmonic analysis, Fourier restrictions.
• Generalization of classical Fourier analysis to locally compact Abelian groups: Haar measure, convolution, dual groups and the Fourier transform, positive-definite functions, inversion theorem, Plancherel's theorem, Pontryagin duality theorem, Bohr compactification.
• Time-frequency analysis and frame theory: Frames and Riesz bases in Hilbert spaces, reproducing kernel Hilbert spaces. Wavelet transform, short-time Fourier transform, wavelet and Gabor frames. Localization operators in time-frequency space, uncertainty principles and spectral properties. Introduction to quantum harmonic analysis.

Learning outcome

(1) Knowledge: The student has in-depth knowledge of key concepts and methods in modern harmonic analysis and time-frequency analysis.

(2) Skills: The student can apply methods from modern harmonic analysis and time-frequency analysis to related areas within mathematics. This includes placing what they learn in the context of both established and new mathematical theory, as well as identifying relevant literature.

(3) Competence: The student can participate in scientific discussions, present results and start to conduct research at an international level in modern and classical harmonic analysis and time-frequency analysis.

Learning methods and activities

The course will be taught when needed, but is offered only as a guided self-study if there are few PhD students enrolled.

The student participates actively in the course through oral presentations of the syllabus throughout the semester.

Course materials

Will be announced at the start of the course.