傅立叶分析导论目录
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The introductory guide to Fourier Analysis begins with an overview of its origins:
Chapter 1, "The Genesis of Fourier Analysis": This chapter explores the foundation of the theory through the concepts of the vibrating string and the heat equation. It includes exercises and problems for readers to apply the theory.
Building upon this, Chapter 2, "Basic Properties of Fourier Series", discusses examples, uniqueness of Fourier series, convolutions, and convergence methods like Cesaro and Abel summability, with exercises and problem sets.
Chapter 3 focuses on the convergence of Fourier Series, examining mean-square convergence and pointwise convergence with accompanying exercises and problems.
Applications of Fourier Series are showcased in Chapter 4, featuring the isoperimetric inequality, Weyl's equidistribution theorem, a non-differentiable function, and the heat equation on the circle, along with exercises.
Chapter 5 delves into the Fourier Transform on R1, covering its elementary theory, applications to differential equations, Poisson summation formula, and the Heisenberg uncertainty principle. Exercises and problems are also provided.
Expanding the scope, Chapter 6 introduces the Fourier Transform on Rd, discussing preliminaries, wave equations, and more advanced topics. Chapter 7 is dedicated to finite Fourier Analysis, while Dirichlet's Theorem is covered in the Appendix.
The guide concludes with notes, references, a bibliography, and a glossary of symbols for a comprehensive understanding of the subject.
