Can the limitations of the Riemann integral be overcome? What is its relationship with modern analysis?
The theory of Lebesgue integration is a crucial component in the development of modern analysis. This book is an in-depth real analysis textbook, which introduces the basic theory of modern analysis and the basic skills of analysis. Based on the knowledge of real analysis, the theory of interpolation of operators and the Fourier transform theory are further introduced systematically. The main contents include: abstract measures and integrals, measure and topology, Lebesgue integration on Rn, the interpolation of operators on Lp(Rn), Hardy-Littlewood maximal function, convolution and the Fourier transform. They play an important role in harmonic analysis, partial differential equations, probability and numerical analysis. This book is moderately difficult and detailed, focusing on the combination of abstract and concrete, and training readers to skillfully use modern analysis.
This textbook is an excellent reference book for readers studying the fields of Harmonic analysis and partial differential equations. It is intended for advanced undergraduate and graduate students in university mathematics, as well as mathematicians and physicists in general.
Contents:
- Abstract Measure and Integral
- Measure and Topology
- Lebesgue Integration on ℝn
- The Interpolation of Operator on Lp(ℝn)
- Hardy–Littlewood Maximal Function
- Convolution
- The Fourier Transform
Readership: Advanced undergraduate and graduate students, mathematicians and physicists.
Professor Shanzhen Lu, born in November 1939, is a well-known mathematician and educator. He graduated from East China Normal University in July 1961, began teaching in the Department of Mathematics of Beijing Normal University in China in September at the same year. From 1980 to 1982, he was a visiting scholar at Washington University in St. Louis, where he was appointed as a professor of mathematics. He was promoted from lecturer to professor at Beijing Normal University in May 1983, approved by the Academic Degrees Committee of the State Council of the People's Republic of China as a doctoral supervisor in January 1984. He has successively served as the editorial board member of Science China, Chinese Science Bulletin, Acta Mathematica Sinica, Frontiers of Mathematics in China and a number of foreign mathematical journals, deputy editor of Advances in Mathematics, and editor-in-chief of Pure Mathematics. In addition, he has served as the chairman of major Harmonic analysis conferences at home and abroad.Professor Lu has long been engaged in the research of Harmonic analysis, and has made important contributions to the development of this field, and many praised research results by international peer experts during his academic career of more than 50 years. Among them, the most prominent work is his research work on the Bochner-Riesz means at critical index, which has a wide impact on this field. Professor Lu and his collaborators created and made full use of the block space theory to completely solve R Fefferman's conjecture about the almost everywhere convergence of the Bochner-Riesz means at critical index, which was cited by famous mathematicians such as E M Stein, Y Meyer and others. Besides, Professor Lu proved the localized convergence theorem for the q-order strong summation of Bochner-Riesz means at critical index, which R Fefferman commented: this problem has not been solved for 36 years since the work of S Bochner and K Chandrasekharan in 1948. Tao's articles on the almost convergence of the Bochner-Riesz means also highlights Professor Lu's research work about it. In addition, Professor Lu and his students studied oscillatory singular integrals with rough kernels, and gave a discriminant criterion about their boundedness in Lp spaces. Professor Lu is an excellent mathematics educator who has trained many outstanding students, such as Professor Heping Liu of Peking University in China, Professor Dachun Yang and Professor Yong Ding of Beijing Normal University, Professor Dunyan Yan of the University of Chinese Academy of Sciences, and Professor Xiaochun Li of the University of Illinois at Urbana-Champaign, etc., who have made important achievements in their respective fields.Professor Kunyang Wang, national model of teaching in China, graduated from the Department of Mathematical Mechanics of Peking University in China in 1966. In 1978, he was admitted to Beijing Normal University, where he completed his master's and doctoral studies, under the tutelage of Professor Yongsheng Sun. He went to Germany, Australia, Canada and other countries as a senior visiting scholar or visiting professor for many times. Professor Wang has served as a member of the Beijing Municipal Committee of the Chinese People's Political Consultative Conference, a member of the Mathematics Subcommittee of the Higher Education Mathematics and Statistics Teaching Committee of the Ministry of Education of China, the director of the Education Work Committee of the Chinese Mathematical Society, the editorial board member of journals such as Advances in Mathematics (China) and Journal of Mathematical Research and Exposition (China), and has participated in the reform of national teaching projects for many times. Professor Wang has published more than 60 academic papers, two academic monographs and three textbooks. The Chinese textbooks he compiled, such as Real Analysis, Concise Mathematical Analysis, Special Research on Mathematical Analysis, Lecture Notes on Mathematical Analysis, Mathematical Analysis, Orthogonal Series, etc., have been deeply preferred by teachers and students. He has also successively published 8 teaching reform papers, hosted and completed the key project of teaching reform of the Teacher Normal Department of the Ministry of Education of China, the famous course project created by the National Science Base in China, and the excellent course project of Mathematical Analysis. He was rated as an advanced worker by the Ministry of Education and the National Natural Science Foundation of China, and won the Beijing Famous Teacher Award.
