Manfred Stoll - Böcker

This classic textbook has been used successfully by instructors and students for nearly three decades. This timely new edition offers minimal yet notable changes while retaining all the elements, presentation, and accessible exposition of previous editions. A list of updates is found in the Preface to this edition.This text is based on the author’s experience in teaching graduate courses and the minimal requirements for successful graduate study. The text is understandable to the typical student enrolled in the course, taking into consideration the variations in abilities, background, and motivation. Chapters one through six have been written to be accessible to the average student,w hile at the same time challenging the more talented student through the exercises.Chapters seven through ten assume the students have achieved some level of expertise in the subject. In these chapters, the theorems, examples, and exercises require greater sophistication and mathematical maturity for full understanding.In addition to the standard topics the text includes topics that are not always included in comparable texts. Chapter 6 contains a section on the Riemann-Stieltjes integral and a proof of Lebesgue’s t heorem providing necessary and sufficient conditions for Riemann integrability. Chapter 7 also includes a section on square summable sequences and a brief introduction to normed linear spaces. C hapter 8 contains a proof of the Weierstrass approximation theorem using the method ofaapproximate identities. The inclusion of Fourier series in the text allows the student to gain some exposure to this important subject. The final chapter includes a detailed treatment of Lebesgue measure and the Lebesgue integral, using inner and outer measure. The exercises at the end of each section reinforce the concepts. Notes provide historical comments or discuss additional topics.

This classic textbook has been used successfully by instructors and students for nearly three decades. This timely new edition offers minimal yet notable changes while retaining all the elements, presentation, and accessible exposition of previous editions. A list of updates is found in the Preface to this edition.This text is based on the author’s experience in teaching graduate courses and the minimal requirements for successful graduate study. The text is understandable to the typical student enrolled in the course, taking into consideration the variations in abilities, background, and motivation. Chapters one through six have been written to be accessible to the average student,w hile at the same time challenging the more talented student through the exercises.Chapters seven through ten assume the students have achieved some level of expertise in the subject. In these chapters, the theorems, examples, and exercises require greater sophistication and mathematical maturity for full understanding.In addition to the standard topics the text includes topics that are not always included in comparable texts. Chapter 6 contains a section on the Riemann-Stieltjes integral and a proof of Lebesgue’s t heorem providing necessary and sufficient conditions for Riemann integrability. Chapter 7 also includes a section on square summable sequences and a brief introduction to normed linear spaces. C hapter 8 contains a proof of the Weierstrass approximation theorem using the method ofaapproximate identities. The inclusion of Fourier series in the text allows the student to gain some exposure to this important subject. The final chapter includes a detailed treatment of Lebesgue measure and the Lebesgue integral, using inner and outer measure. The exercises at the end of each section reinforce the concepts. Notes provide historical comments or discuss additional topics.

This monograph provides an introduction and a survey of recent results in potential theory with respect to the Laplace-Beltrami operator D in several complex variables, with special emphasis on the unit ball in Cn. Topics covered include Poisson-Szegö integrals on the ball, the Green's function for D and the Riesz decomposition theorem for invariant subharmonic functions. The extension to the ball of the classical Fatou theorem on non-tangible limits of Poisson integrals, and Littlewood's theorem on the existence of radial limits of subharmonic functions are covered in detail. The monograph also contains recent results on admissible and tangential boundary limits of Green potentials, and Lp inequalities for the invariant gradient of Green potentials. Applications of some of the results to Hp spaces, and weighted Bergman and Dirichlet spaces of invariant harmonic functions are included. The notes are self-contained, and should be accessible to anyone with some basic knowledge of several complex variables.

This comprehensive monograph is ideal for established researchers in the field and also graduate students who wish to learn more about the subject. The text is made accessible to a broad audience as it does not require any knowledge of Lie groups and only a limited knowledge of differential geometry. The author's primary emphasis is on potential theory on the hyperbolic ball, but many other relevant results for the hyperbolic upper half-space are included both in the text and in the end-of-chapter exercises. These exercises expand on the topics covered in the chapter and involve routine computations and inequalities not included in the text. The book also includes some open problems, which may be a source for potential research projects.