Advanced complex analysis
The main purpose of the course is to give a brief introduction to spaces of analytic functions and their applications. Being interested and important by itself this topic is at the crossroad of complex analysis and operator theory, and provides a powerful machinery in numerous applications both in pure and applied mathematics.
Lecturer
Date
10th September ~ 5th December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 09:50 - 11:25 | A3-2-303 | ZOOM 07 | 559 700 6085 | BIMSA |
Syllabus
The tentative plan of the course is as follows:
(1) Short reminder of basic facts of the standard course of complex analysis (if need be).
(2) Harmonic functions in the unit disk
(3) Boundary behaviour of analytic functions.
(4) Jenssen formula. Blaschke products.
(5) Classes Hp: boundary behavior, inner-outer factorization.
(6) Carleson interpolation theorem (without proof)
(7) Hilbert transform, harmonic conjugate
(8) Reminder on conformal mappings (if need be)
(9) Hp spaces in the halfplane
(10) Paley-Wiener theorem
(11) Titchmarsh convolution theorem
(12) Phragmen Lindelof theorem
(13) Survey on Paley-Wiener spaces (with partial proofs)
(14) Interpolation in Paley-Wiener spaces.
If possible I would like to reserve some time to cover some topics on student’s choice (if any). The possible topics are
• Functions of bounded mean oscillations and Hilbert transform;
• Hardy-Littlewood maximal function and its application to functions in the unit circle;
• Harmonic measure;
• Application of conformal mappings;
• Bargmann-Fock spaces and applications;
• Something else.
The lectures will be completed with home exercises. I also plan to give several topics for students work.
(1) Short reminder of basic facts of the standard course of complex analysis (if need be).
(2) Harmonic functions in the unit disk
(3) Boundary behaviour of analytic functions.
(4) Jenssen formula. Blaschke products.
(5) Classes Hp: boundary behavior, inner-outer factorization.
(6) Carleson interpolation theorem (without proof)
(7) Hilbert transform, harmonic conjugate
(8) Reminder on conformal mappings (if need be)
(9) Hp spaces in the halfplane
(10) Paley-Wiener theorem
(11) Titchmarsh convolution theorem
(12) Phragmen Lindelof theorem
(13) Survey on Paley-Wiener spaces (with partial proofs)
(14) Interpolation in Paley-Wiener spaces.
If possible I would like to reserve some time to cover some topics on student’s choice (if any). The possible topics are
• Functions of bounded mean oscillations and Hilbert transform;
• Hardy-Littlewood maximal function and its application to functions in the unit circle;
• Harmonic measure;
• Application of conformal mappings;
• Bargmann-Fock spaces and applications;
• Something else.
The lectures will be completed with home exercises. I also plan to give several topics for students work.
Reference
1. P. Koosis, Introduction to Hp spaces, 1998,
2. B. Ya. Levin, Lectures on entire functions, 1996
2. B. Ya. Levin, Lectures on entire functions, 1996
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Yurii Lyubarskii received PhD degree in mathematics in 1974 in the Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences and Doctor degree in St. Petersburg branch of Russian mathematical institute in 1990. He taught at the Norwegian University of Science and Technology and also at St. Petersburg State University. Scientific interests of Yu. Lyubarskii include complex and harmonic analysis and applications to the signal analysis.