Prerequisites: MA543 Functional Analysis or Equivalent Course
Preamble / Objectives (Optional): This course is intended to learn advanced analysis on Hardy spaces, which can be used for further study and in many research areas, like, uncertainty principle, Schrodinger equation, PDE, image processing, signal processing, etc.
Course Content/ Syllabus: Invariant Subspaces of L^2(μ): Doubly Invariant Subspaces, Simply Invariant Subspaces, Inner Functions, Uniqueness Theorem, Invariant Subspaces of L^2(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 L^2(T; μ): Generalized Fourier Series, Bases of Exponentials in L^2(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.
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