Course Category：Advanced Course
Aims & Requirements:
The goal is to study the inter-relationship between complex analysis, harmonic analysis and signal analysis, including higher dimensional cases. In the latter, one naturally uses Clifford algebra and Clifford analysis, that is a modern and fruitful research area whose beauty and applications now are being explored by the world mathematicians. The lectures will include some material on the latest development in signal analysis.
As prerequisite knowledge we require basic knowledge in real and complex analysis. Some fundamental knowledge in harmonic analysis will have great advantage.
Ch.1. Boundary Value of Cauchy Integrals on the Line and on the Circle
Contents: Plemelj formulas, Hilbert transform and circular Hilbert transform on the line and on the circle, respectively, relevant knowledge in Fourier transform and Fourier series
Ch.. 2. Hardy Spaces Theory
Contents: Hardy spaces in the complex analysis setting, boundary value characterization of Hardy spaces, spectral characterization of Hardy spaces
Ch. 3. Analytic Signals and Mono-components
Contents: Analytic signals, instantaneous amplitude and frequency, mono-components, adaptive decomposition of signals, results in relation to Nevanlinna class
Ch. 4. Clifford Analysis
Contents: Basic concepts of Clifford algebra and Clifford analysis, Dirac operator, monogenic functions, Cauchy’s theorem and Cauchy’s formula
Ch. 5. Singular Integral Operators
Contents: The singular integral operators that form an operator algebra equivalent to the Fourier multiplier operators induced by bounded analysis functions in sectors in the complex plane, equivalence to the Cauchy-Dunford functional calculus of bounded analytic functions in sectors
Ch.6. Cauchy Singular Operator vs. Hilbert transforms
Contents: Inner and outer Hilbert transforms on curves and surfaces in general, inner and outer Schwarz kernels, inner and outer Hilbert transforms on the unit sphere
(i) The first three chapters of J.B. Garnet’s book: Bounded Analytic Functions, Academic Press, 1987.
(ii) W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.
(iii) E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, 1987.