调和分析讲义
Harmonic Analysis Lecture Notes
摘要：
这些笔记提出了第一个研究生课程在调和分析。第一部分强调傅里叶级数，因为调和分析的许多方面已经在经典的背景下出现。希尔伯特变换是在圆上处理的，例如，它被用来证明傅里叶级数的L^p收敛。极大函数和卡尔德龙-齐格蒙德分解在R^d中进行了处理，以便在课程的第二部分（研究傅里叶变换）中再次应用。课程的最后一部分处理带限函数，泊松求和和不确定性原则。附录中涵盖了分布函数和插值。每章开头的参考文献为希望更深入研究或更广泛地漫游该主题的学生提供指导。
英文摘要：
These notes present a first graduate course in harmonic analysis. The first part emphasizes Fourier series, since so many aspects of harmonic analysis arise already in that classical context. The Hilbert transform is treated on the circle, for example, where it is used to prove L^p convergence of Fourier series. Maximal functions and Calderon--Zygmund decompositions are treated in R^d, so that they can be applied again in the second part of the course, where the Fourier transform is studied. The final part of the course treats band limited functions, Poisson summation and uncertainty principles. Distribution functions and interpolation are covered in the Appendices. The references at the beginning of each chapter provide guidance to students who wish to delve more deeply, or roam more widely, in the subject.
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