Advanced Topics in Complex Analysis
A.Y. 2021/2022
Learning objectives
Introduction to the more known spaces of holomorphic functions in the disc and in a half plane. Analysis of their properties, with attention to the proofs techniques. The focus is on Hardy and (weighted) Bergman spaces on the disc and in a half plane, Paley-Wiener spaces and Bernstein spaces.
Expected learning outcomes
Knowledge of the topics and results, and application to exercises that need also computational techniques.
Lesson period: Second semester
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Specific information on the delivery modes of the teaching activities for the academic year 2021/22 will be provided during the coming months, depending on the evolution of the public health.
Course syllabus
Hardy spaces Hp(D) on the unit disc. Function spaces with reproducing kernel. Bergman spaces Ap(D) and weighted Bergman spaces Aνp(D). Lp boundedness of the Bergman and Cauchy-Szego projections. Fourier tranform on R of the spaces L1 and L2. Paley—Wiener theorems. Bergman and Hardy spaces in the upper half plane. Spaces of entire functions: Paley—Wiener, Fock and de Branges spaces. Introduction to several complex variables theory. Holomorphic fuctions on the unit disc, Hardy and Bergman spaces, projections and boundedness in Lp.
Prerequisites for admission
Fundamental prerequisite is the course Analisi Complessa. Moreocer, the good knowledge of the courses corsi Analisi Reale and Analisi di Fourier is strongly suggested.
Teaching methods
Classroom lessons with the use of a blackboard.
Teaching Resources
- P. Duren, A. Schuster, Bergman Spaces, Mathematical Survey and Monographs v. 100, American Mathematial Society, Providence 2004.
- K. Hoffman, Banach Spaces of Analytic Functions, Dover, New York 1988.
- R. Paley, N. Wiener Fourier Transforms in the Complex Domain, Colloquium Publications v. 19 American Mathematial Society, Providence 2000.
- M. M. Peloso Appunti del corso.
- K. Hoffman, Banach Spaces of Analytic Functions, Dover, New York 1988.
- R. Paley, N. Wiener Fourier Transforms in the Complex Domain, Colloquium Publications v. 19 American Mathematial Society, Providence 2000.
- M. M. Peloso Appunti del corso.
Assessment methods and Criteria
The final examination consists of an oral exam.
- In the oral exam, the student will be required to illustrate concepts, examples and results presented during the course and will be required to solve problems quite similar at those presented in the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
- In the oral exam, the student will be required to illustrate concepts, examples and results presented during the course and will be required to solve problems quite similar at those presented in the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor:
Peloso Marco Maria
Educational website(s)
Professor(s)
Reception:
By appointment
My office, room 1021 Dipartimento di Matematica