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

This textbook provides a coherent, integrated look at various topics from undergraduate analysis. It begins with Fourier series, continues with Hilbert spaces, discusses the Fourier transform on the real line, and then turns to the heart of the book, geometric considerations. This chapter includes complex differential forms, geometric inequalities from one and several complex variables, and includes some of the author's original results. The concept of orthogonality weaves the material into a coherent whole. This textbook will be a useful resource for upper-undergraduate students who intend to continue with mathematics, graduate students interested in analysis, and researchers interested in some basic aspects of Cauchy-Riemann (CR) geometry. The inclusion of several hundred exercises makes this book suitable for a capstone undergraduate Honors class.​This second edition contains a significant amount of new material, including a new chapter dedicated to the CR geometry of the unit sphere. This chapter builds upon the first edition by presenting recent results about groups associated with CR sphere maps. **From reviews of the first edition:**__The present book developed from the teaching experiences of the author in several honors courses. .... All the topics are motivated very nicely, and there are many exercises, which make the book ideal for a first-year graduate course on the subject. .... The style is concise, always very neat, and proofs are given with full details. Hence, I certainly suggest this nice textbook to anyone interested in the subject, even for self-study.__ **Fabio Nicola, Politecnico di Torino, Mathematical Reviews**__D’Angelo has written an eminently readable book, including excellent explanations of pretty nasty stuff for even the more gifted upper division players .... It certainly succeeds in hooking the present browser: I like this book a great deal.__ **Michael Berg, Loyola Marymount University, Mathematical Association of America** 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 This textbook provides a coherent, integrated look at various topics from undergraduate analysis. It begins with Fourier series, continues with Hilbert spaces, discusses the Fourier transform on the real line, and then turns to the heart of the book, geometric considerations. This chapter includes complex differential forms, geometric inequalities from one and several complex variables, and includes some of the author's original results. The concept of orthogonality weaves the material into a coherent whole. This textbook will be a useful resource for upper-undergraduate students who intend to continue with mathematics, graduate students interested in analysis, and researchers interested in some basic aspects of Cauchy-Riemann (CR) geometry. The inclusion of several hundred exercises makes this book suitable for a capstone undergraduate Honors class.​ This second edition contains a significant amount of new material, including a new chapter dedicated to the CR geometry of the unit sphere. This chapter builds upon the first edition by presenting recent results about groups associated with CR sphere maps. From reviews of the first edition: The present book developed from the teaching experiences of the author in several honors courses. .... All the topics are motivated very nicely, and there are many exercises, which make the book ideal for a first-year graduate course on the subject. .... The style is concise, always very neat, and proofs are given with full details. Hence, I certainly suggest this nice textbook to anyone interested in the subject, even for self-study. Fabio Nicola, Politecnico di Torino, Mathematical Reviews D’Angelo has written an eminently readable book, including excellent explanations of pretty nasty stuff for even the more gifted upper division players .... It certainly succeeds in hooking the present browser: I like this book a great deal. Michael Berg, Loyola Marymount University, Mathematical Association of America Hermitian Analysis: From Fourier Series to Cauchy-Riemann Geometry provides a coherent, integrated look at various topics from analysis. It begins with Fourier series, continues with Hilbert spaces, discusses the Fourier transform on the real line, and then turns to the heart of the book: geometric considerations in several complex variables. The final chapter includes complex differential forms, geometric inequalities from one and several complex variables, finite unitary groups, proper mappings, and naturally leads to the Cauchy-Riemann geometry of the unit sphere. The book thus takes the reader from the unit circle to the unit sphere. This textbook will be a useful resource for upper-undergraduate students who intend to continue with mathematics, graduate students interested in analysis, and researchers interested in some basic aspects of CR Geometry. It will also be useful for students in physics and engineering, as it includes topics in harmonic analysis arising in these subjects. The inclusion of an appendix and more than 270 exercises makes this book suitable for a capstone undergraduate Honors class--