Mathematical analysis III (MAT903)

The course is aimed at PhD students in complex and harmonic analysis and will consist of three blocks, each corresponding to 5 credit points. The first block is part of a general mathematical education and is compulsory for all the analysis PhD students, while the second part of the course is on more specific subjects and to be chosen from the second and third blocks.

Description: The course is aimed at PhD students in complex and harmonic analysis and will consist of three blocks, each corresponding to 5 credit points. The first block is part of a general mathematical education and is compulsory for all the analysis PhD students, while the second part of the course is on more specific subjects and to be chosen from the second and third blocks.

Literature: W. Rudin, Real and Complex Analysis; T. Ransford, Potential Theory in the Complex Plane; A. Olevskii and A. Ulanovskii, Functions with Disconnected Spectrum; M. Klimek, Pluripotential Theory.

After finishing the course, the student will have knowledge of measure theory and integration, as well as basics of potential theory. In addition, the student will learn either main ideas of Fourier analysis, including sampling and interpolation of band-limited functions (option 1), or of basics on holomorphic functions of several variables, complex manifolds and pluripotential theory.

Module 1 (5ECTS- FIXED): Measure theory, integration and potential theory

Contents: general measure theory and Lebesgue integration; basics of potential theory in the complex plane and Rn.

Module 2 (5ECTS - option1): Fourier analysis

Contents: Fourier transform; functional spaces; sampling and interpolation of band-limited functions.

Module 3 (5ECTS - option2): Several complex variables and pluripotential theory

Contents: basics on holomorphic functions of several variables; complex manifolds; introduction to pluripotential theory.