Advanced Topics in Complex Analysis
A.Y. 2026/2027
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.
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: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
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
First semester
Course syllabus
Hardy spaces Hp(D) on the unit disc.
Inner-outer factorization. Singular inner functions.
Shift operator and invariant subspaces, Beurling's theorem.
Wold's theorem. Discussion of the invariant subspace problem.
Function spaces with reproducing kernel.
Bergman spaces Ap(D) on the unit disc.
Lp boundedness of the Bergman and Cauchy-Szego projections.
Fourier tranform on R of the spaces L1 and L2.
Paley—Wiener theorems.
Hardy spaces in the upper half plane. Introduction to the de Branges spaces.
Inner-outer factorization. Singular inner functions.
Shift operator and invariant subspaces, Beurling's theorem.
Wold's theorem. Discussion of the invariant subspace problem.
Function spaces with reproducing kernel.
Bergman spaces Ap(D) on the unit disc.
Lp boundedness of the Bergman and Cauchy-Szego projections.
Fourier tranform on R of the spaces L1 and L2.
Paley—Wiener theorems.
Hardy spaces in the upper half plane. Introduction to the de Branges spaces.
Prerequisites for admission
Analisi Complessa, Analisi di Fourier.
Teaching methods
Standard blackboard lectures. Lecture notes available online.
Teaching Resources
· Lecture notes of the course.
· N. Nikolski, Functions, and Systems_ An Easy Reading. Hardy, Hankel, and Toeplitz. vol.1, American Mathematial Society, Providence 2002.
· K. Hoffman, Banach Spaces of Analytic Functions, Dover, New York 1988.
· P. Duren, A. Schuster, Bergman Spaces, Mathematical Survey and Monographs v. 100, American Mathematial Society, Providence 2004.
· Y. Katznelson, An Introduction to Harmonic Analysis, Dover 2nd edition, New York 1976.
· R. Paley, N. Wiener Fourier Transforms in the Complex Domain, Colloquium Publications v. 19 American Mathematial Society, Providence 2000.
· N. Nikolski, Functions, and Systems_ An Easy Reading. Hardy, Hankel, and Toeplitz. vol.1, American Mathematial Society, Providence 2002.
· K. Hoffman, Banach Spaces of Analytic Functions, Dover, New York 1988.
· P. Duren, A. Schuster, Bergman Spaces, Mathematical Survey and Monographs v. 100, American Mathematial Society, Providence 2004.
· Y. Katznelson, An Introduction to Harmonic Analysis, Dover 2nd edition, New York 1976.
· R. Paley, N. Wiener Fourier Transforms in the Complex Domain, Colloquium Publications v. 19 American Mathematial Society, Providence 2000.
Assessment methods and Criteria
Final oral exam.
Goals. 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 Bergman spaces on the disc and in a half plane, Paley-Wiener spaces and Bernstein spaces.
Expected outcome. Knowledge of the topics and results, and application to exercises that need also computational techniques.
Goals. 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 Bergman spaces on the disc and in a half plane, Paley-Wiener spaces and Bernstein spaces.
Expected outcome. Knowledge of the topics and results, and application to exercises that need also computational techniques.
MATH-03/A - Mathematical Analysis - University credits: 6
Lessons: 42 hours
Professor:
Peloso Marco Maria
Professor(s)
Reception:
By appointment
My office, room 1021 Dipartimento di Matematica