Hermitian Analysis: From Fourier Series to Cauchy-Riemann Geometry (Cornerstones)

Preface to the second edition
	Preface to the first edition
Contents
1 Introduction to Fourier series
	1 Introduction
	2 A famous series
	3 Trigonometric polynomials
	4 Constant coefficient differential equations
	5 The wave equation for a vibrating string
	6 Solving the wave equation via exponentiation
	7 Integrals
	8 Approximate identities
	9 Definition of Fourier series
	10 Summability methods
	11 The Laplace equation
	12 Uniqueness of Fourier coefficients for continuous functions
	13 Inequalities
2 Hilbert spaces
	1 Introduction
	2 Norms and inner products
	3 Subspaces and linear maps
	4 Orthogonality
	5 Orthonormal expansion
	6 Polarization
	7 Adjoints and unitary operators
	8 A return to Fourier series
	9 Bernstein\'s theorem
	10 Compact Hermitian operators
	11 Sturm-Liouville theory
	12 Generating functions and orthonormal systems
	13 Spherical harmonics
3 Fourier transform on R
	1 Introduction
	2 The Fourier transform on the Schwartz space
	3 The dual space
	4 Convolutions
	5 Plancherel theorem
	6 Heisenberg uncertainty principle
	7 Differential equations
	8 Pseudo-differential operators
	9 Hermite polynomials
	10 More on Sobolev spaces
	11 Inequalities
4 Geometric considerations
	1 The isoperimetric inequality
	2 Elementary L2 inequalities
	3 Unitary groups
	4 Proper mappings
	5 The derivative as a linear map
	6 Complex differential forms and vector fields
	7 Differential forms of higher degree
	8 Pullbacks and integrals
	9 Volumes of parametrized sets
	10 Volume computations
	11 Inequalities
	12 Unifying remarks
5 The unit sphere and CR geometry
	1 Geometry of the unit sphere
	2 Positivity conditions for Hermitian polynomials
	3 Groups associated with holomorphic mappings
	4 Maps between balls
	5 Examples of unitary invariance
	6 Behavior of Γf under various constructions
	7 A criterion for a power series being a polynomial
	8 A criterion for a formal power series to be a rational function
6 Appendix
	1 The real and complex number systems
	2 Metric spaces
	3 Integrals
	4 Exponentials and trig functions
	5 Complex analytic functions
	6 Probability
Notation used in this book
References
Index