Continuity, integration and Fourier theory
This book is a textbook for graduate or advanced undergraduate students in mathematics and mathematical physics. It is devided in eight chapters: 1. The space of continuous functions. 2. Theorems of Korovkin and Stone- Weierstrass, 3. Fourier series of continuous functions, 4. Integration and differentiation, 5. Spaces \(L_ p\) and convolutions, 6. Fourier series of summable functions, 7. Fourier integral, 8. Additional results. The first two chapters are devoted to real or complex continuous functions on \({\mathbb{R}}^ k\) and the third to the elementary theory of Fourier series. Chapters 4 and 5 deal with integration theory of Lebesgue measure in Euclidean space as well as for abstract measures. The presentation is brief but with complete proofs. The topics of chapters 6 and 7 are the Riesz-Fisher theorem and the Plancherel theorem, respectively. The last chapter is concerned with various applications, for instance: Functions of analytic type, the Hausdorff-Young theorem, the Heat equation. The book contains more than 60 adequate exercises and ends with a list of references and subject index.
scientific article; zbMATH DE number 42074