MA 650: Advanced Course on Hardy spaces (3-1-0-8), January to May, 2022
A brief perspective about the course:
This course is intended to learn advanced analysis on Hardy spaces, which can be used for further study and in many research areas, like, harmonic analysis, operator theory, uncertainty principle, Schrodinger equation, PDE, image processing, signal processing, etc.
Syllabus:
Invariant Subspaces of L2(μ): Doubly Invariant Subspaces, Simply
Invariant Subspaces, Inner Functions, Uniqueness Theorem, Invariant Subspaces of
L2(μ). Applications: The Problem of Weighted Polynomial
Approximation, A Probabilistic Interpretation, The Inner-Outer Factorization,
Arithmetic of Inner Functions, A Characterization of Outer Functions, Fourier
Series, Szego Infimum and The Riesz Brothers' Theorem. Hp Classes and
Canonical Factorization: Identifying Hp(D) and Hp(T),
Jensen's Formula and Jensen's Inequality, The Boundary Uniqueness Theorem,
Blaschke Products, Non-tangential Boundary Limits, the Riesz-Smirnov Canonical
Factorization, Approximation by Inner Functions and Blaschke Products. Szego
infimum, and Phragmen-Lindelof Principle: Szego infimum and Weighted Polynomial
Approximation, Recognize an Outer Function, Locally Outer Function, The Smirnov
Class N+, a Conformally Invariant Framework, The Generalized
Phragmen-Lindelof Principle. Harmonic Analysis on L2(T;
μ):
Generalized Fourier Series, Bases of Exponentials in L2(T;
μ),
Harmonic Conjugates, the Hilbert Transform, the Helson-Szego Problem. Hp spaces
Upper Half-Plane: A Unitary Mapping from Lp(T) to Lp(R),
Cauchy Kernels and Fourier Transform, The Hardy Space Hp+
= Hp(C+), Canonical Factorization and other relevant
properties as compared to disc, Invariant Subspaces.
Textbooks/ References:
1. N. Nikolskii, Operators, Functions, and Systems: an Easy
Reading, vols. I and II., AMS, Providence, RI, 2002
2. P. Koosis, Introduction to Hp Spaces, Cambridge University Press, Cambridge,
1980.
3. M. Rosenblum, J. Rovnyak, Topics in Hardy classes and univalent functions.
BirkhauserVerlag, Basel ,1994
4. J. Garnett, Bounded Analytic Functions, Springer, New York, 2007
5. E. Fricain and J. Mashreghi, The theory of H(b) spaces. Vol. 1. New
Mathematical Monographs, 20. Cambridge University Press, Cambridge, 2016.
6. E. Fricain and J. Mashreghi, The theory of H(b) spaces. Vol. 2. New
Mathematical Monographs, 21. Cambridge University Press, Cambridge, 2016.
Class slot - B, Exam slot - B, Class Room- MS Team:
Classes will be held on Mon, Thu, Fri (10:00 -10:55) and Wed (08:00- 08:55)
If you are willing to credit/audit the course, please join the MS team though the link : Grp_MA650
Assignments: Assignment 1, Assignment 2, Assignment 3, Assignment 4, Assignment 5
Lecture notes: Complete file
Exams: Quiz-I, MidSem, Quiz-II, EndSem, Viva-Voce
Number of student attended: Four