At the beginning of the "Cultural Revolution", Cheng Minde was severely impacted, and then he was quarantined for seven years. He worked in Jiangxi cadre school for two years. During this period, he was always full of confidence in the party's socialist cause. Once conditions permit, he began to resume scientific research. From 65438 to 0973, he began to study Walsh transform and its application in image band compression according to the actual situation at that time, organized interdisciplinary seminars and engaged in information processing research. He is one of the pioneers and advocates of pattern recognition and image processing research in China.
1976 After the downfall of the Gang of Four, Cheng Minde was completely liberated politically. 1978 as the first director of Peking University Institute of Mathematics; 1980 was elected as an academician of China Academy of Sciences; 1982 to 1986 as the chairman of Beijing Mathematical Society; 1983- 1988 served as vice chairman of chinese mathematical society. During this period, he made a series of organizational work for the mathematics department of Peking University, the mathematics research institute and the national mathematics development, and achieved remarkable results. He used to be one of the leaders of the leading group of applied mathematics of the State Education Commission, a member of the mathematics evaluation group of the the State Council Academic Degrees Committee, the deputy director of the editorial board of the national textbook, the editorial board of China Science and Science Bulletin, and the leader of the academic leading group of the Tianyuan Fund for Mathematics of the National Foundation. He is still the editor-in-chief of Modern Mathematics Basic Series, the editor-in-chief of Peking University Mathematics Series, and the deputy editor-in-chief of Mathematical Yearbook and Journal of Applied Mathematics. The first basic problem of Fourier series is what condition the function f(x) satisfies, and its Fourier series converges to f(x0) at x0. In 1872, P.D.G. Dubois-Raymond constructs a counterexample, which shows that the continuity of function in x0 cannot guarantee the convergence of Fourier series in X.. So people adopted a new concept of convergence-summation method. The simplest summation method is (c, 1) summation, that is, consider the limit of the arithmetic mean of the sum of the first n parts of the series when n→∞. Fei Jie (1900) proved that as long as the function is continuous at x0, its Fourier series can sum to f(x0) at x0 (c, 1). It can be seen that the concept of summation is more suitable for Fourier series theory than the concept of convergence. Cheng Minde's early work was to study various summation methods and summation factors of unit Fourier series.
Another problem of Fourier series theory is uniqueness. The formulation of this question is, if a trigonometric series converges (or can be summed) to an integrable function, can it be asserted that this trigonometric series must be the Fourier series of this function? Or in a narrow sense, if a trigonometric series converges (or can be summed) to 0, can it be asserted that the coefficients of this trigonometric series are all 0? Both G.F.B. Riemann and Cantor have made great achievements in the study of uniqueness of unary trigonometric series, which promoted the generation of point set theory.
Until the forties of this century, the theory of harmonic analysis, including the above basic problems, was relatively complete only for univariate functions. Because of the difficulty in principle, multivariate harmonic analysis has not made an essential breakthrough. In 1930s and 1940s, due to the need of studying partial differential equations, harmonic analysts have been seeking progress in this field. In the late 1940s, Cheng Minde adapted to this trend, and his research direction shifted from monistic harmonic analysis to multivariate analysis, starting with the uniqueness theory of multiple trigonometric series, and achieved important results.
In order to prove the uniqueness theorem of multiple trigonometric series, Cheng Minde developed an independent field, that is, the study of biharmonic functions. It is known that harmonic function is a quadratic continuously differentiable function satisfying Laplace equation △u=0. M- biharmonic function is a 2m-degree continuously differentiable function, which satisfies the equation △mu=0. The problem is how to describe the m- biharmonic of U when only the smoothness of U is known (for example, only 2m-2 continuously differentiable is known). German blaschke solved this problem in 19 16. In 1930s, when D. Nicholson studied the uniqueness of the general M. Cheng Minde, he found that the conditions he gave were only necessary conditions, not sufficient conditions. He introduced the generalized multiple Laplacian operation in 1950 (recorded as? M), and it is proved that △mu=0 if and only if △ mu = 0 under the condition that u is 2m-2 continuously differentiable. mu=0 .
Since the 1950s, multivariate harmonic analysis has made great progress, and one of the topics is the study of fractional integral. The fractional integral of multivariate function in the whole N-dimensional Euclidean space was introduced by M. Riesz in 1949, which is the Riesz potential. For periodic functions or finite regions, there is no obvious similarity. Cheng Minde and Chen Yonghe defined the fractional integral and fractional Laplacian operation of periodic function through Bochner-Riess average of multiple Fourier series, and studied their properties and their relationship with Polev space in detail. Due to the need of embedding theorem, in 1950s, many people in the Soviet Union and the United States studied the fractional integrals of periodic functions and functions defined in finite regions. Among these works, Cheng Minde and Chen Yonghe were published in Journal of Peking University 1957 and 1959 respectively, and published in Abstract of Polish Academy of Sciences 1956 respectively.
The breakthrough of multivariate harmonic analysis is internationally recognized as the basic work of A.P. Calderón and A.Zygmund on singular integral operators in 1952. The vigorous development in the future formed a complete theory of multivariate harmonic analysis. Cheng Minde noticed this progress as early as 1950s, and organized a seminar in Peking University on 1962 to learn the theory of singular integral operators. After the Cultural Revolution, he quickly resumed the research work of multivariate harmonic analysis, translated E. Stein's singular integrals and differentiability of functions, and personally taught graduate students. Four doctors and nearly 20 masters have been trained in this field. The research team led by him has been active in the international frontier of multivariate harmonic analysis. They have made outstanding achievements in Hardy space, Besov space, singular integral operator, Hankel operator and so on, and have been highly praised by their international counterparts. He and his students have compiled their lectures for graduate students into a book "Practical Analysis" and published it. Function approximation theory is a branch of mathematics developed at the beginning of this century. Its basic idea is to use simple functions (such as polynomials or trigonometric polynomials) to approximate complex functions with poor properties, which is of great significance in theory and practical application. Before 1950s, approximation theory mainly studied the approximation problem of univariate functions. The approximation of multivariate function has made great progress since 1950s. One of the most common methods to approximate multivariate periodic functions is to use the summation method of cyclic sums of Fourier series-δ-order Bochner-Riess average.
This summation method, the greater the δ, the better the performance. δ has a critical exponent δ0= 1/2, which is the demarcation number to describe this summation method. 1947, two Indian mathematicians proved that large δ(δ>:δ0 10 a), using SδR to approximate Lipschitz function of α order can achieve an ideal approximation degree, but the results are obviously inaccurate. From 65438 to 0956, Cheng Minde first studied the theory of multivariate triangular approximation in China. He cooperated with Chen Yonghe to completely solve the problem above the critical order (δ >). Approximation of Bochner-Reese mean. They proved that as long as δ >; Δ 0, the ideal approximation degree can be achieved. They also linked the concept of fractional integral of periodic function with the theory of multivariate trigonometric approximation, and obtained rich results. These results are not only included in the monograph because of the integrity of its system, but also have great influence on the theory of multivariate triangular approximation. Until 1980s, on the basis of Cheng Minde's work, the research on Bochner-Riess average equal to or less than the critical order was still a very active topic. In China, some mathematicians continue to work in this direction. In addition, because Fourier series is closely related to mathematical physics, the research results of Cheng Minde and others have been used in the numerical analysis of partial differential equations by Guo Benyu and others. From 1973, Cheng Minde began to study pattern recognition and image processing from high-dimensional Walsh transform. Walsh transform is another orthogonal expansion similar to Fourier expansion. In many cases, it is more suitable for the analysis of digital radio signals than Fourier transform. In 1970s, the application of two-dimensional Walsh transform in TV band compression was successful in computer simulation and laboratory experiments. But in theory, even in one-dimensional case, there is still a lack of systematic and complete work. 1978, Cheng Minde systematically and completely analyzed the high-dimensional Walsh transform, proved the convergence theorem and sampling theorem, and demonstrated the superiority of Walsh transform in digital image frequency band compression. In cooperation with students, I completed the first monograph on pattern recognition, Introduction to Image Recognition.
Because of the application of computer, the research of pattern recognition and image processing developed very rapidly in the 1960s and 1970s, but it started late in China. Cheng Minde not only engaged in theoretical research, but also further established the information mathematics major of the Department of Mathematics of Peking University, leading everyone to study fingerprint identification, geographic information database and visual simulation. He, Shi Qingyun and their graduate students made great discoveries in fingerprint identification, thus developing a new generation of high-functional and practical automatic fingerprint identification system, which entered the international market on 1990 and made contributions to China's economic development. Based on the scientific research group led by Cheng Minde, Peking University has successively established the Interdisciplinary Information Science Center and the State Key Laboratory of Visual and Auditory Information Processing, with Cheng Minde as the director of the academic committees of the center and laboratory.
Cheng Minde adheres to the principle of paying equal attention to mathematical theory and practice in academic thought. He attaches great importance to the independent development of mathematical theory, and thinks that all mathematical research should not have practical background, and should also attach great importance to the application of mathematics. In the 1980s, when some people doubted whether a mathematician should engage in pattern recognition, he insisted on the research direction of pattern recognition. It is under the guidance of his correct thinking that the information science major of Peking University Mathematics Department and Peking University Information Center have made great progress. 1952 the department of mathematics and mechanics in Peking University is facing a great development situation. The number of students has increased rapidly from dozens to thousands, and the specialty of mechanics and computational mathematics has increased from a single mathematics specialty. However, the lack of teachers can not meet the requirements of development, and teaching is facing the task of reform. As one of the main leaders of the teaching and research section and department, Cheng Minde started with strengthening the teaching of basic courses and vigorously built various majors. He personally taught more than 200 students specialized courses in mathematical analysis, and cultivated students with extremely rigorous analytical style, thus establishing a fine tradition of attaching importance to basic training in the newly-built Department of Mathematical Mechanics in Peking University. When the teaching quality gradually stabilized, in 1955, together with young teachers such as Lin Jianxiang and Ding, he put forward suggestions on actively developing scientific research in colleges and universities in time. In addition, at that time, the Department of Mathematical Mechanics of Peking University was merged from the Mathematics Departments of Peking University, Tsinghua and yenching university, and the teachers came from different units. Cheng Minde, together with Duan Xuefu, then head of the department, got the support of Professor Jiang Zehan and Professor Xu Xianyu under the leadership of the Party organization, gave full play to the role of teachers in the first three schools, trusted young teachers and strengthened their training, and paid attention to establishing a United and harmonious atmosphere and a positive and rigorous style of study, which made the new department form a fine fashion. This fashion played an extremely important role in the later development of the Department of Mathematics of Peking University.
After the "Cultural Revolution", the mathematics departments of Peking University and China, which experienced ten years of catastrophe, are facing the situation of recovery and redevelopment. Cheng Minde actively supports ideological rectification. It has consolidated and developed the applied mathematics specialty and information science specialty of the Department of Mathematics of Peking University, and signed several major scientific research project agreements. After the establishment of Peking University Institute of Mathematics, he served as the first director, creating a good research environment and a good atmosphere for free discussion in the institute. He took various measures to help a large number of young and middle-aged people grow up quickly. At home, he first resumed the theoretical research of Peking University's multivariate harmonic analysis from 65438 to 0977. Then at 1978, under his active initiative, function theory, as a theoretical discipline, resumed its academic activities at the earliest in China. He overcame many difficulties and successfully assisted Professor Wu Wenjun in organizing and convening the first international academic conference on differential equations and differential geometry initiated by internationally renowned mathematician Mr. Chen Shengshen in 1980, which set a high-standard example for the international exchange of mathematics in China and had a far-reaching impact on improving the level of mathematics in China. Later on 1984, he presided over the international seminar on analytical science, 1985 organized the international conference on approximation theory, and 1988 presided over the academic activities of harmonic analysis in the Institute of Mathematics of Nankai University. He did a lot of practical work for chinese mathematical society to return to the World Mathematical Union. He strongly supported the establishment and activities of the Institute of Mathematics of Nankai University, participated in and led the National Summer Teaching Center for Mathematics Postgraduates initiated by Mr. Chen Shengshen to the State Education Commission, which provided good conditions for improving the modern mathematics level of mathematics postgraduates in China. He also made great contributions to Sino-US cooperation in training graduate students. 1985, Cheng Minde and Xu Lizhi jointly founded the international English mathematics magazine Approximation Theory and Its Application and served as the editor-in-chief.
After the "Cultural Revolution", China's mathematics field showed a prosperous scene, and many young people stood out and made excellent works at home and abroad. At this time, Mr. Chen Shengshen suggested that China could catch up with the world advanced level in mathematics in the early 20th century and build China into a great mathematical country in the 20th century. In order to achieve this goal, Cheng Minde and others, with the support of the State Science and Technology Commission, the National Fund Committee and the State Education Commission, held the first academic seminar "2 1.988 Century China's Mathematics View" in Nankai University. There are 122 people in China and 45 people abroad, many of whom are young people who are studying or have obtained doctoral degrees. Under the auspices of Cheng Minde, Hu, Wu Wenjun and others, the meeting discussed the development of mathematics in China. The meeting won a fund from the Ministry of Finance for the development of mathematics in China-Mathematics Tianyuan Fund. The academic leading group of Tianyuan Fund headed by Cheng Minde decided to use it to support a number of key projects, especially young people, and create conditions for their development. At the same time, we decided to support the photocopying of mathematics books and periodicals and the translation and publication of mathematics books and materials, and try our best to improve the conditions of mathematics research in some countries. 1990 "the second 2 1 century China mathematics prospect conference" was held in Nankai university, and everyone was determined to achieve the goal of catching up with the world advanced level in mathematics through solid work. The meeting presented a new atmosphere of unity and struggle.
When he was young, Cheng Minde was taciturn and inarticulate. At a party he attended while studying in the United States, his tutor Botshner introduced him as a "silent mathematician". After returning home, it was the tide of history that rushed him to the position of administrative leader. Due to historical reasons, mathematics in China is naturally divided into two activity centers, the North and the South. Cheng Minde studied and worked in the south in his youth, and then taught in the north for a long time. When studying in the United States, he came into contact with many internationally renowned mathematicians. This objectively provides favorable conditions for his work. But more importantly, he never put personal gains and losses first, and always put the overall situation first. He is generous to others, always considerate of others and strict with himself. He is strong-willed. No matter what difficulties he meets, he always asks himself to be down-to-earth and even work in obscurity until he reaches his goal. He treats people sincerely and never says anything against his will, so he can unite people and give full play to their roles. Academically, he is not conservative, and has always encouraged young people to create and even encourage young people to surpass themselves. These are the reasons why he can contribute to the development of mathematics in China and gain people's trust and respect.