Rudi Weikard

Measure and Integration: Examples, Concepts, and Applications offers a comprehensive introduction to the fundamental principles and methods of real analysis, providing readers with a solid foundation through clear explanations, rigorous proofs, and an abundance of thoughtfully constructed exercises. From the very first chapter, students are encouraged to engage actively with concepts, applying them to a range of practical examples that reinforce both understanding and analytical skills. The book’s structured approach ensures that readers not only grasp key theorems and core techniques, but also develop the problem-solving abilities essential for higher-level mathematics and related applications. The text also delves into advanced areas such as integration on product spaces, Radon functionals, functions of bounded variation, Lebesgue-Stieltjes measures, convolutions, probability, and differential equations. Each chapter concludes with advanced exercises, clearly marked for difficulty, allowing both students and instructors to tailor their study or coursework. An appendix with complete solutions supports independent learning, making the book a valuable resource for both classroom use and self-study.

This graduate textbook offers an introduction to the spectral theory of ordinary differential equations, focusing on Sturm–Liouville equations. Sturm–Liouville theory has applications in partial differential equations and mathematical physics. Examples include classical PDEs such as the heat and wave equations. Written by leading experts, this book provides a modern, systematic treatment of the theory. The main topics are the spectral theory and eigenfunction expansions for Sturm–Liouville equations, as well as scattering theory and inverse spectral theory. It is the first book offering a complete account of the left-definite theory for Sturm–Liouville equations. The modest prerequisites for this book are basic one-variable real analysis, linear algebra, as well as an introductory course in complex analysis. More advanced background required in some parts of the book is completely covered in the appendices. With exercises in each chapter, the book is suitable for advancedundergraduate and graduate courses, either as an introduction to spectral theory in Hilbert space, or to the spectral theory of ordinary differential equations. Advanced topics such as the left-definite theory and the Camassa–Holm equation, as well as bibliographical notes, make the book a valuable reference for experts.